University  of  California  •  Berkeley 

THE  THEODORE  P.  HILL  COLLECTION 

of 

EARLY  AMERICAN  MATHEMATICS  BOOKS 


J,      -y— •• 
i4jj£&^3^ 

/7 


0-1 


&c\ 

YOwV \V.WvVV^-  ,V    »>-.  ,v\> 


RECOMMENDATIONS 


THOMSON'S  ABRIDGMENT  OF  DAY'S  ALGEBRA, 


THE  great  excellence  of  DAY'S  ALGEBRA  has  been  so  fully  acknowl- 
edged by  the  public,  through  a  period  of  nearly  thirty  years,  that  it 
would  be  superfluous  to  accompany  it  with  any  formal  recommendation. 
:  All  that  instructors  will  require  to  be  assured  is,  that  the  present 
Abridgment,  by  Mr.  James  B.  Thomson,  faithfully  presents  the  spirit 
and  character  of  the  original.  I  have  examined  it  sufficiently  to  feel 
satisfied  that  such  is  the  fact ;  that  while  it  presents  to  the  young  learner 
the  science  in  a  simple  and  attractive  form,  it  surpasses  most  similar 
treatises  in  the  aptness  of  its  illustrations,  the  accuracy  of  its  defini- 
tions, and  the  value  and  copiousness  of  its  principles. 

DENISON  OLMSTED. 

YALE  COLLEGE,  June  6,  1843. 

Speaking  of  the  Abridgment  of  Day's  Algebra,  Prof.  Silliman  says, 
I  have  full  confidence  in  the  views  expressed  above  bv  Prof  Olm- 
sted." 

We  fully  concur  in  the  above  recommendation  of  the  Abridgment 
of  Day's  Algebra,  from  Prof.  Olmsted. 

Prof.  A.  D.  STANLEY,  Yale  College. 
J.  L.  KINGSLEY, 
C.  A.  GOODRICH, 
T.  D.  WOOLSEY, 

C.  U.  SHEPARD, 
T.  A.  THATCHER, 

Tutor  J.  NOONEY, 

D.  POWERS, 
L.  J.  DUDLEY, 
P.  K.  CLARK, 
HAWLEY  OLMSTED, 

Principal  of  Hopkins'  Grammar  School. 

A.  N.  SKINNER, 

Principal  of  Select  Classical  School  for  boys. 
L.  A.  DAGGETT, 
Principal  of  Select  School  for  boys. 


NEW  HAVEN,  June  12th,  1843. 

Messrs.  DURRIE  &  PECK,— I  have  examined,  by  request,  Thomsons' 
Abridgment  of  Day's  Algebra,  and  take  pleasure  in  commending  it 
*to  teachers,  as  having  all  the  well  known  merits  of  the  original,  with 
such  additional  illustrations  and  exercises  as  adapt  it  to  the  capacities 
of  the  young,  and  the  method  of  instruction  in  schools  and  acadamies. 
The  fact  that  this  Abridgment  has  received  the  approbation  of  Pres. 
Day  himself,  is  a  sufficient  proof  of  its  character. 

Respectfully  yours,  STILES  FRENCH, 

Principal  of  the  Collegiate  arid  Commercial  School,  at  Wooster  Place. 

I  have  examined,  with  pleasure,  the  Abridgment  of  Day's  Algebra, 
by  James  B.  Thomson,  A.  M.  The  accuracy  of  definition  and  clear- 
ness of  illustration,  which  characterize  the  large  work,  are  faithfully 
preserved  in  this ;  and  the  addition  of  a  large  number  of  examples, 
and  some  changes  in  the  arrangement,  greatly  enhance  the  value  of 
the  production.  I  am  acquainted  with  no  elementary  work  on  the  sub- 
ject so  well  adapted  to  the  purposes  of  instruction,  as  this  Abridg- 
ment, and  shall  gladly  introduce  it  into  my  school. 

DANIEL  D.  TOMPKINS  M'LAUGHLIN, 
Principal  of  a  Classical  School  in  New  York  city. 

NEW  YORK,  June  12th,  1843. 

Mr.  J.  B.  THOMSON — DEAR  SIR, — The  merits  of  the  Algebra,  of 
which  yours  is  an  Abridgment,  and  my  confidence  in  your  knowledge 
of  what  was  needed  to  adapt  it  to  the  use  of  schools  and  academies, 
and  in  your  ability  for  the  task  you  undertook,  led  me  to  expect  that 
you  would  make  the  work  what  I  find  it  to  be.  one  eminently  adapted 
to  the  wants  of  that  large  class  of  pupils  in  our  schools,  whose  age 
and  circumstances  require  a  treatise  more  full  in  explanation,  better 
furnished  with  examples  for  practice,  and  yet  more  limited  in  extent, 
than  those  in  general  use.  Yours  truly, 

WM.  H.  RUSSELL, 
Principal  of  New  Haven  Family  School  for 'boys. 

From  the  New  York  Tribune,  June  12,  1843. 

Day's  Algebra  is  by  far  the  best  work  for  beginners  that  has  ever 
been  published:  and  this  Abridgment  serves  to  adapt  it  still  more  per- 
fectly and  completely  to  the  wants  and  necessities  of  the  young.  The 
elements  of  mathematics  are  very  clearly  and  plainly  developed ;  the 
definitions  are  simple  and  comprehensive,  and  the  problems  well  adap- 
ted .  to  illustrate  the  principles  taught.  We  commend  this  Abridg- 
ment to  the  attention  of  teachers.  There  is  no  reason  why  Algebra 
should  not  be  much  more  generally,  taught  than  it  is  at  present ;  it  is 
eminently' useful,  and  this  work  will  greatly  facilitate  its  introduction 
even  into  our  common  schools  throughout  the  country. 


ELEMENTS 


ALGEBRA,        1 


BEING    AN 


ABRIDGMENT   OF   DAY'S   ALGEBRA, 


ADAPTED    TO    THE 


CAPACITIES     OF     THE     YOUNG, 

AND    THE 

METHOD    OF   INSTRUCTION, 

IN 

SCHOOLS   AND  ACADEMIES. 

BY 

JAMES  B.   THOMSON,  A.  M. 


NEW   HAVEN: 

DURRIE   &  PECK. 

PHILADELPHI  A — S  MITH    &   PECK. 

NEW    Y  o  R K — R OBINSON,   PRATT  &  Co. 

B  O  S  T  O  N — C  ROCKER    &   BREWSTER. 

1843. 


Entered  according  to  Act  of  Congress,  in  the  year  1843, 

by  JEREMIAH  DAY  and  JAMES  B.  THOMSON, 
in  the  Clerk's  Office  of  the  District  Court  of  Connecticut. 


N.  B.  The  Key  to  this  work  will  shortly  be  published  for 
the  use  of  teachers. 

An  abridgment  of  LEGENDRE'S  Geometry,  by  the  same  au- 
thor, will  also  soon  be  published  for  the  use  of  schools  and 
academies. 


Printed  by  B.  L.  Hamlen. 


PREFACE. 


PUBLIC  opinion  has  pronounced  the  study  of  Algebra  to  be 
a  desirable  and  important  branch  of  popular  education.  This 
decision  is  one  of  the  clearest  proofs  of  an  onward  and  sub- 
stantial progress  in  the  cause  of  intellectual  improvement  in 
our  country.  A  knowledge  of  algebra  may  not  indeed  be 
regarded  as  strictly  necessary  to  the  discharge  of  the  common 
duties  of  life ;  nevertheless  no  young  person  at  the  present 
•day  is  considered  as  having  a  "  finished  education"  without 
an  acquaintance  with  its  rudiments. 

The  question  with  parents  is,  not  "  how  little  learning  and 
discipline  their  children  can  get  through  the  world  with  ;" 
but,  "  how  much  does  their  highest  usefulness  require  ;"  and 
"what  are  the  lest  means  to  secure  this  end?" 

It  has  long  been  a  prevalent  sentiment  among  teachers  and 
the  friends  of  education,  that  an  abridgment  of  Day's  Alge- 
bra, adapted  to  the  wants  of  schools  and  academies,  would 
greatly  facilitate  this  object.  Whilst  his  system  has  been 
deemed  superior  to  any  other  work  before  the  public,  and 
most  happily  adapted  to  the  circumstances  of  college  students, 
for  whom  it  was  especially  prepared ;  it  has  also  been  felt, 
that  a  smaller  and  cheaper  werk,  combining  the  simplicity  of 
language  and  the  uyirivaled  clearness  with  which  the  princi- 
ples of  the  science  are  there  stated,  would  answer  every  pur- 
pose for  beginners,  and  at  the  same  time  bring  the  subject 
within  the  means  of  the  humblest  child  in  the  land. 


IV  PREFACE. 

In  accordance  with  this  sentiment,  such  a  work  has  been 
prepared,  and  is  now  presented  to  the  public.  The  design 
of  the  work,  is  to  furnish  an  easy  and  lucid  transition  from 
the  study  of  arithmetic  to  the  higher  branches  of  algebra  and 
mathematics,  and  thus  to  subserve  the  important  interests  of 
a  practical  and  thorough  education. 

Its  arrangement,  with  but  few  exceptions,  is  the  same  as 
that  of  the  large  work.  For  the  sake  of  more  convenient 
reference,  the  division  by  compound  divisors,  and  the  bino- 
mial theorem,  both  of  which  were  originally  placed  after 
mathematical  infinity,  are  brought  forward,  the  former  being 
placed  after  division  of  simple  quantities,  and  the  latter  after 
involution  of  simple  quantities.  The  reason  for  deferring 
the  consideration  of  compound  division  in  the  original,  was 
the  fact  that  some  of  the  terms  contain  powers  which  it  is 
impossible  for  pupils  at  this  stage  of  their  progress  to  under- 
stand. To  avoid  this  difficulty  in  the  present  work,  whenever 
a  power  occurs,  instead  of  using  an  index  before  it  has  been 
explained,  the  letter  is  repeated  as  a  factor  in  the  same  man- 
ner as  in  multiplication,  and  also  in  dividing  by  a  simple 
quantity.  (Arts.  80,  94.)  Afterwards,  under  division  of 
powers,  copious  examples  of  dividing  by  compound  quantities 
which  have  indices,  are  given. 

As  continued  arithmetical  proportion  and  arithmetical  pro- 
gression are  one  and  the  same  thing,  they  are  placed  con- 
tiguously in  the  same  section.  For  the  same  reason  continu- 
ed geometrical  proportion  and  geometrical  progression  are 
placed  in  a  similar'  manner.  Mathematical  infinity,  roots  of 
binomial  surds,  infinite  series.,  indeterminate  co-efficients, 
composition  and  resolution  of  the  higher  equations,  with  equa- 
tions of  curves,  are  subjects  which  belong  to  the  higher  and 
more  difficult  parts  of  algebra,  and  it  has  been  thought  ad- 
visable to  omit  them  in  the  abridgment.  Those  who  have 


PREFACE.  Y 

leisure  and  are  desirous  of  acquiring  a  knowledge  of  these" 
subjects,  will  find  them  explained  with  all  the  author's  accus* 
tomed  clearness  and  ability  in  his  large  work,  to  which  they 
are  respectfully  referred.  The  similarity  between  the  ope-* 
rations  in  addition,  subtraction,  multiplication  and  division  of 
radical  quantities,  and  those  of  the  same  rules  in  powers  \ 
also  between  involution  and  evolution  of  radicals,  and  of  pow-* 
ers,  has  been  more  fully  developed,  and  the  rules  of  both  are 
expressed  in  as  nearly  the  same  language  as  the  nature  of 
the  case  would  admit.  It  has  also  been  attempted  to  illus- 
trate the  "Binomial  Theorem,"  on  the  principles  of  induc- 
tion ;  the  second  method  of  completing  the  square  in  quad* 
ratic  equations  has  been  demonstrated  ;  and  other  methods 
of  completing  the  square  pointed  out,  which,  so  far  as  the 
author  knows,  are  original. 

It  was  a  cardinal  point  with  the  distinguished  author  of  the 
large  work,  never  to  use  one  principle  in  the  explanation  of 
another,  until  it  had  itself  been  explained,  a  characteristic  of 
rare  excellence  in  school-books  and  works  of  science.  This 
plan  has  been  rigidly  adhered  to,  in  the  preparation  of  the 
abridgment.  After  the  principles  have  been  separately  ex- 
plained, and  illustrated  by  examples,  they  have  then  been 
carefully  summed  up  in  the  present  work,  and  placed  in  the 
form  of  a  general  rule.  This,  it  is  thought  by  competent 
judges,  will  be  found  very  convenient  and  useful  both  to 
teachers  and  scholars.  By  this  means  the  peculiar  advanta- 
ges of  the  inductive  and  synthetic  modes  of  reasoning  have 
been  united,  and  made  subservient  alike  to  the  pleasure  and 
facility  both  of  imparting  and  acquiring  knowledge. 

As  a  guide  to  the  attention  of  beginners  to  the  more  im« 
portant  principles  of  the  science,  a  few  practical  questions 
are  placed  at  the  foot  of  the  page.  They  are  intended  to  be 
merely  suggestive.  No  thorough  teacher  will  confine  himself 

1* 


VI  PREFACE. 

to  the  questions  of  an  author,  however  full  and  appropriate 
they  may  be.  From  a  conviction  that  the  answers  to  prob- 
lems have  a  tendency  to  destroy  rather  than  promote  habits 
of  independent  thinking  and  reasoning  in  the  minds  of  learn- 
ers, they  have  nearly  all  been  excluded  from  the  book.  For 
the  convenience  of  teachers  and  others,  who  may  entertain 
different  views  upon  this  point,  the  answers  are  given  in  a 
Key,  in  which  may  also  be  found  a  statement  and  solution  of 
the  more  difficult  examples  contained  in  the  work. 

The  formation  of  correct  habits  of  study  and  of  thought, 
together  with  the  extermination  or  prevention  of  lad  ones, 
.  requires  the  utmost  vigilance  and  skill  on  the  part  of  teachers. 
They  must  insist  upon  thoroughness,  upon  "  the  why  and 
ivherefore"  of  each  successive  step,  or  in  most  cases,  their  pu- 
pils will  fall  into  superficial  and  mechanical  habits,  which  are 
equally  destructive  of  high  attainments  and  future  usefulness. 
To  mold  the  youthful  mind  right,  is  an  arduous  and  responsi- 
ble task ;  sufficient  to  crush  the  jaded  spirit  and  shattered 
nerves  of  a  poorly  paid  teacher.  Nevertheless  it  is  a  high 
and  noble,  as  well  as  indispensable  work.  Every  conscien- 
tious teacher  therefore,  who  appreciates  the  importance  of  his 
profession,  or  is  worthy  to  be  entrusted  with  this  responsible 
charge,  will  cheerfully  devote  his  energies  to  the  work,  what- 
ever may  be  the  sacrifice,  or  resign  his  trust  to  more  faithful 
and  able  hands.  "  In  mathematics  as  in  war,  it  should  be  made 
a  principle,"  says  the  author  of  the  large  work,  "  not  to  advance, 
while  any  thing  is  left  unconquered  behind.  Neither  is  it  suf- 
ficient that  the  student  understands  the  nature  of  the  proposi- 
tion, or  method  of  operation,  before  proceeding  to  another. 
He  ought  also  to  make  himself  familiar  with  every  step,  by  a 
careful  attention  to  the  examples."  It  is  emphatically  true 
in  algebra,  that  "  practice  makes  perfect."  For  this  reason, 
the  number  of  problems  in  the  present  work,  has  been  nearly 


PREFACE.  Vll 

doubled ;  the  most  of  those  added  are  original,  and  are  calcu- 
lated to  make  the  principles  of  the  science  more  familiar. 

The  merits  of  DAY'S  ALGEBRA,  are  too  well  known  and  ap- 
preciated to  require  any  comment.  The  fact  that  it  has  been 
adopted,  as  a  text-book,  by  so  many  of  our  colleges  and  higher 
seminaries  of  learning ;  that  during  the  last  fourteen  years 
more  than  forty  large  editions  have  been  called  for,  affords 
sufficient  evidence  of  the  superior  rank,  which  it  holds  in  pub- 
lic estimation. 

With  regard  to  the  abridgment,  it  is  fervently  hoped,  that 
all  who  have  felt  the  want  of  a  lucid  introductory  work  upon 
this  subject,  will  here  find  the  fulfillment  of  their  wishes. 
Those  teachers  who  have  used  the  large  work  in  their  colle- 
giate course  or  elsewhere,  and  who  may  have  occasion  to  use 
this,  will  at  least  be  saved  from  the  inconvenience  of  unlearn- 
ing one  set  of  rules,  and  of  learning  a  new  and  perhaps  an 
inferior  set,  a  work  by  no  means  unfrequent,  and  of  no  small 
magnitude  and  perplexity.  On  the  other  hand,  those  schol- 
ars, who  chance  to  use  the  abridgment  in  their  preparatory 
course,  will  avoid  the  necessity  of  unlearning  its  rules  and 
modes  of  operation  in  algebra,  should  they  have  occasion  to 
use  the  large  work  in  the  subsequent  part  of  their  education. 

It  has  been  the  endeavor  of  the  author  to  divest  the  study 
of  algebra,  once  so  formidable,  of  all  its  intricacy  and  repul- 
siveness ;  to  illustrate  its  elementary  principles  so  clearly, 
that  any  school-boy  of  ordinary  capacity,  may  understand 
and  apply  them ;  and  thus  to  render  this  interesting  and  use- 
ful science,  more  attractive  to  the  young.  With  what  suc- 
cess these  efforts  have  been  attended,  it  remains  for  his  fellow 
teachers  and  an  impartial  public  to  decide. 

J.  B.  THOMSON. 
New  Haven,  May  20,  1843. 


NOTICE. 


HAVING  been  myself  prevented,  by  impaired  health,  and 
official  engagements,  from  preparing  an  abridgment  of  my 
Introduction  to  Algebra,  I  applied  to  Mr.  J.  B.  THOMSON, 
to  abridge  the  work,  in  such  a  manner  as  to  adapt  it  to  the 
demand  and  use  of  the  higher  schools  and  academies. 

I  had  confidence  in  his  mathematical  talents  and  attain- 
ments, and  his  practical  knowledge,  derived  from  several 
years'  experience  in  teaching  algebra,  as  qualifying  him  to 
make  the  abridgment  proposed ;  and  I  am  gratified  to  find, 
on  examination,  that  our  design  has  been  skilfully  and  satis- 
factorily executed.  The  abridgment,  it  is  hoped,  will  be  fa- 
vorably received,  by  those  who  approve  of  the  original  work. 

J.  DAY. 

Yale  College,  May  29,  1843. 


CONTENTS. 


SECTION  I. 

Page. 

Introduction,  -  -  13 
Algebra  defined  and  illustrated,  -  13,14 
Algebraic  Notation  by  letters  and  signs  explained,  -  -  15-20 
Operations  stated  in  common  language,  translated  into  algebraic 

language,  -  -21 

Translation  of  algebraic  operations  into  common  language,  22 

Positive  and  Negative  Quantities,       ...  23 

Axioms,     -                                                              -       .  ._*j-:       "  25 

SECTION  II. 

Addition  illustrated  and  defined,          *            -                         4*  .'  26 

Adding  quantities  which  are  alike  and  have  like  signs,  -  27 

Adding  quantities  which  are  alike,  but  have  unlike  signs,     -  28 

General  Rule  of  Addition,                                                             "  *  '  29 

SECTION  III. 

Subtraction  illustrated,  &c.,     -  31 

General  Rule  of  Subtraction,        -                                                    -  32 

Proof,  33 

m 

SECTION  IV. 

Multiplication  illustrated,  &c.,     -            -            -                          -  35 
Rule  for  the  signs  of  the  Product,  39 
The  reason  why  the  product  of  two  negatives  should  be  affirma- 
tive, illustrated,             -  -                                                      -             -  41 
General  Rule  of  Multiplication,          ....  43 


CONTENTS. 


SECTION  V. 

Page. 

Division  illustrated,  &c.,  -  -  44, 45 

Signs  of  the  Quotient,    ---...  47 

Division  by  Compound  Divisors,  -  -         49 

General  Rule  of  Division,       .....  51 


SECTION  VI. 

Algebraic  Fractions  explained,  &c.,        -  -  53, 54 

The  eifect  of  the  signs  placed  before  the  numerator  and  denomi- 
nator, also  before  the  dividing  line  of  a  fraction,  with  their 
changes,  &c.,  -  ....  55^  55 

Reduction  of  Fractions,     -  -  -  57-60 

Addition  of  Fractions,  .....  61 

Subtraction  of  Fractions,  -  -  -  -        63 

Multiplication  of  Fractions,     .....  65 

Division  of  Fractions,        ......        68 

SECTION  VII. 

Simple  Equations  illustrated,  &c.,       ....  71 

Reduction  of  Equations  by  Transposition,  •  -  -73 

Reduction  of  Equations  by  Multiplication,     ...  76 

Clearing  of  Fractions,  &c.,  -----         76 

Reduction  of  Equations  by  Division,  removing  co-efficients,  &c.,       78 
Converting  a  proportion  into  an  equation,  and  an  equation  into 

a  proportion,  -  -  -  *..  -  -        79,80 

Substitution  illustrated,  &c.,         -  -  -  -        80 

General  Rule  for  solving  Simple  Equations,  83 

• 
SECTION  VIII. 

Involution,  powers  of  different  degrees,  &c.,  •            -            -91 

Indices,  Direct  and  Reciprocal  Powers,          -  -            -              92 

To  involve  a  quantity  to  any  required  power,  -            -        .?  *•       93 

To  involve  a  fraction,               -        :.»/'.'        -  -          '-*><:           95 

To  involve  a  binomial,  or  residual  quantity,  -        ':•*,•-*&        -        97 


CONTENTS.  XI 

Page. 

Binomial  Theorem  illustrated,                         -  98 

General  Rule  for  involving  binomials  to  any  required  power,     -  101 

Addition  of  Powers,                             ...  105 

Subtraction  of  Powers,     -                          -                                       -  106 

Multiplication  of  Powers,        -                          ...  JQ7 

Division  of  Powers,            ......  1Q9 

Examples  of  Compound  Divisors  with  Indices,  111 

Greatest  Common  Measure,                        -                                       -  112 

Fractions  containing  Powers,               -                         -  114 


SECTION  IX. 

Roots  illustrated,  &c.,       -  -        ^  -  115 

Powers  of  Roots,          -  118 

Evolution,  the  Rule,  &c.  -  -  -  120 

The  Root  of  a  Fraction,  -  .  ...  122 

Signs  to  be  placed  before  roots,     .....  122 

:    Reduction  of  Radical  Quantities,        ....  124 

:"  Addition  of  Radical  Quantities,    -  -  -  129 

The  similarity  of  this  and  the  five  following  rules  to  the  same 

rules  in  Powers,  -  -        "A"^  129 

Subtraction  of  Radical  Quantities,      ....  131 

Multiplication  of  Radical  Quantities,       -  -  -  132 

Division  of  Radical  Quantities,  ...  137 

Involution  of  Radical  Quantities,  -  140 

Evolution  of  Radical  Quantities,        ....  142 


SECTION  X. 

Reduction  of  Equations*by  Involution,     ....       144 

Reduction  of  Equations  by  Evolution,            ...  145 

Quadratic  Equations,  Pure  and  Affected,              -            -  150,  151 

First  method  of  Completing  the  Square,         ...  153 

Second  Method  of  Completing  the  Square,          -  •       156 

Demonstration  of  the  second  method,  &c.,     ...  157 

Other  methods  of  Completing  the  Square  pointed  out,     •  158,159 

General  Rule  for  reducing  Quadratic  Equations,        -            •  163 


Xll  CONTENTS. 


SECTION  XL 

Two  Unknown  Quantities, 

To  Exterminate  one  of  two  unknown  quantities  by  Comparison, 

To  Exterminate,  &c.,  by  Substitution,     - 

To  Exterminate,  &c.,  by  Addition  and  Subtraction, 

Three  Unknown  Quantities, 

Four  or  more  Unknown  Quantities, 

SECTION  XII. 

Ratio  and  Proportion,        -  ... 

Different  kinds  of  Ratio,         ... 
Proportion,  its  different  kinds,  &c., 

SECTION  XIII. 

Arithmetical  Proportion  and  Progression, 

Various  rules,  or  formulas  in  Arithmetical  Progression, 

SECTION  XIV. 

Geometrical  Proportion  and  Progression, 

Case  1. — Changes  in  the  Order  of  the  Terms, 

Case  2. — Multiplying,  or  Dividing  by  the  Same  Quantity,     - 

Case  3. — Comparing  one  Proportion  with  another, 

Case  4. — Addition  and  Subtraction  of  equal  Ratios,  - 

Case  5. — Compounding  Proportions,      „:?.;,. 

Case  6. — Involution  and  Evolution  of  the  Terms,     - 

Continued  Geometrical  Proportion  or  Progression, 

Various  rules,  or  formulas  in  Geometrical  Progression, 

SECTION  XV.      * 

Evolution  of  Compound  Quantities, 

Extraction  of  the  Square  Root  of  Compound  Quantities, 

SECTION  XVI. 

Application  of  Algebra  to  Geometry,       .... 
Miscellaneous  Problems,  ..... 


\ 


ALGEBRA. 


SECTION    I. 

INTRODUCTI ON. 

ART.  1.  ALGEBRA  is  a  general  method  of  solving  problems, 
and  of  investigating  the  relations  of  quantities  ly  means  of 
letters  and  sign's. 

I  LLU  STRATI  ON. 

PROB.  1.  Suppose  a  man  divided  72  dollars  among  his 
three  sons  in  the  following  manner :  To  A  he  gave  a  certain 
number  of  dollars  ;  to  B  he  gave  three  times  as  many  as  to 
A ;  and  to  C  he  gave  the  remainder,  which  was  half  as  many 
dollars  as  A  and  B  received.  How  many  dollars  did  he  give 

I  to  each  ? 

1.  To  solve  this  problem  arithmetically,  the  pupil  would 
reason  thus  :  A  had  a  certain  part,  i.  e.  one  share ;  B  receiv- 
ed three  times  as  much,  or  three  shares ;  but  C  had  half  as 
much  as  A  and  B ;  hence  he  must  have  received  two  shares. 
By  adding  their  respective  shares,  the  sum  is  six  shares,  which 
by  the  conditions  of  the  question  is  equal  to  72  dols.  If  then 
6  shares  are  equal  to  72  dols.,  1  share  is  equal  to  %  of  72, 

t;  viz.  12  dols.,  which  is  A's  share.     B  had  three  times  as  many, 
viz.  36  dols.,  and  C  half  as  many  dols.  as  both,  viz.  24  dols. 

QUEST. — What  is  algebra  ?     How  solve  Prob.  1  arithmetically  ? 
2 


14  ALGEBRA.  [Sect.  I. 

2.  Now  to  solve  the  same  problem  by  algebra,  he  would 
use  letters  and  signs ;  thus, 

Let  x  represent  A's  share  ;  then  by  the  conditions, 

o?X3  will  represent  B's  share  ;  and 

4x-±-2  will  represent  C's  share. 

Add  together  the  several  shares,  or  #'s ;  thus,  x-\-3x-\-2x=: 
6x.  Then  will  6x=.72,  for  the  whole  is  equal  to  all  its  parts  ; 
and  la?=12  dols.  A's  share  ;  3^=36  dols.  B's  share  ;  and 
2x=24  dols.  C's  share. 

PROOF.  Add  together  the  number  of  dollars  received  by 
each,  and  the  sum  will  be  equal  to  72,  the  amount  divided. 

In  this  algebraic  solution  it  will  be  observed  ;  First,  that 
we  represent  the  number  of  dollars  which  A  received  by  x. 
Second,  to  obtain  B's  share,  we  must  multiply  A's  share  by  3. 
This  multiplication  is  represented  by  two  lines  crossing  each 
other  like  a  capital  X.  Third,  to  find  C's  share,  we  must 
take  half  the  sum  of  A's  and  B's  share.  This  division  is 
denoted  by  a  line  between  two  dots.  Fourth,  the  addition  of 
their  respective  shares  is  denoted  by  another  cross  formed  by 
a  horizontal  and  perpendicular  line.  Take  another  example. 

PROB.  2.  A  boy  wishes  to  lay  out  96  cents  for  peaches  and 
oranges,  and  wants  to  get  an  equal  number  of  each.  He 
finds  that  he  must  give  2  cents  for  a  peach  and  4  for  an 
orange.  How  many  can  he  buy  of  each  ? 

Let  x  denote  the  number  of  each.  Now  since  the  price  of 
one  peach  is  2  cents,  the  price  of  x  peaches  will  be  x\2 
cents,  or  2x  cents.  For  the  same  reason  xX4,  or  4x  cents 
will  denote  the  price  of  x  oranges.  Then  will  2x-\-4x,  that 
is,  6x,  be  equal  to  96  cents  by  the  conditions,  and  Ix  is  equal 

QUEST. — How  by  algebra  ?  How  denote  A's  share  ?  How  B's  and 
C's  ?  What  is  the  share  of  each  ?  In  Prob.  2,  how  represent  the 
number  of  each  kind  ?  What  represents  the  price  of  each  kind  ? 
The  Ans.  ? 


Arts.  1-7.]  INTRODUCTION.  15 

to  ^  of  96  cents,  viz.  16  cents,  which  is  the  number  he  bought 
of  each. 

2.  QUANTITIES   in   algebra  are   generally  expressed   by 
letters,  as  in  the  preceding  problems.     Thus  b  may  be  put 
for  2,  or  15,  or  any  other  number  which  we  may  wish  to 
express.     It  must  not  be  inferred,  however,  that  the  letter 
used,  has  no  determinate  value.     Its  value  isjixed  for  the  oc- 
casion or  problem  on  which  it  is  employed  ;  and  remains  un- 
altered throughout  the  solution  of  that  problem.     But  on  a 
different  occasion,  or  in  another  problem,  the  same  letter  may 
be  put  for  any  other  number.     Thus  in  Prob.  1,  x  was  put 
for  A's  share  of  the  money.     Its  value  was  12  dols.  and  re- 
mained fixed  through  the  operation.     In  Prob.  2,  x  was  put 
for  the  number  of  each  kind  of  fruit.     Its  value  was  16,  and 
it  remained  so  through  the  calculation. 

3.  By  the  term  quantity,  we  mean  any  thing  which  can  be 
multiplied,  divided,  or  measured.     Thus  a  line,  weight,  time, 
number,  &c.  are  called  quantities. 

4.  The  first  letters  of  the  alphabet  are  used  to  express 
known  quantities  ;  and  the  last  letters,  those  which  are  un- 
known. 

5.  Known  quantities  are  those  whose  values  are  given,  or 
may  be  easily  inferred  from  the  conditions  of  the  problem 
under  consideration. 

6.  Unknown  quantities  are  those  whose  values   are  not 
given. 

7.  Sometimes,  however,  the   quantities,  instead   of  being 
expressed  by  letters,  are  set  down  in  figures. 

QUEST. — How  are  quantities  expressed  in  algebra  ?  What  does 
each  letter  stand  for  ?  Has  the  letter  used  no  determinate  value  ? 
What  is  meant  by  quantity  ?  Give  examples.  Which  letters  are  used 
to  denote  known  quantities  ?  Which  unknown  ?  What  are  known 
quantities  ?  Unknown  ?  Are  figures  ever  used  in  algebra  ? 


16  ALGEBRA.  [Sect.  I. 

8.  Besides  letters  and  figures,  it  will  also  be  seen  that  we 
use  certain  signs  or  characters  in  algebra  to  indicate  the  re- 
lations of  the  quantities,  or  the  operations  which  are  to  be 
performed  with  them,  instead  of  writing  out  these  relations 
and  operations  in  words.     Among  these  is  the  sign  of  addi- 
tion (+),  subtraction  (  — ),  equality  (=),  &c. 

9.  Addition  is  represented  by  two  lines  (+),  one  hori- 
zontal, the  other  perpendicular,  forming  a  cross,  and  is  called 
plus.     It  signifies  "  more,"  or  "  added  to."     Thus  a-\-b  sig- 
nifies that  b  is  to  be  added  to  a.     It  is  read  a  plus  £,  or  a 
added  to  #,  or  a  and  I. 

.  10.  Subtraction  is  represented  by  a  short  horizontal  line 
(_ )  which  is  called  minus.  Thus  a— Z>,  signifies  that  b 
is  to  be  "  subtracted  from"  a ;  and  is  read  a  minus  J,  or  a 
less  b. 

11.  The  sign  -|-  is  prefixed  to  quantities  which  are  consid- 
ered as  positive  or  affirmative  ;  and  the  sign   — ,  to  those 
which  are  supposed  to  be  negative.     For  the  nature  of  this 
distinction,  see  Arts.  36  and  37. 

12.  The  sign  is  generally  omitted  before  ihejlrst  or  leading 
quantity,  unless  it  is  negative ;  then  it  must  always  be  writ- 
ten.    When  no  sign  is  prefixed  to  a  quantity,  -|-  is  always 
understood.     Thus  a-\-b  is  the  same  as  -\-a-\-b. 

13.  Sometimes  both  -f-  and  — ,  (the  latter  being  put  under 
the  former,  -4-,)  are  prefixed  to  the  same  letter.     The  sign  is 
then  said  to  be  ambiguous.     Thus  a-\-b  signifies,  that  in  cer- 

Q.UEST. — What  besides  letters  and  figures  are  used  in  algebra?  What 
is  the  sign  of  addition  ?  How  read  ?  What  does  it  signify  ?  How  is 
subtraction  represented  ?  What  called  ?  What  signify  ?  What  sign 
have  positive  quantities?  What  negative?  What  is  said  as  to  the 
sign  of  the  leading  quantity  ?  When  none  is  expressed,  what  sign  is 
understood  ?  When  both  -J-  and  -  are  prefixed  to  the  same  letter, 
what  is  the  sign  called  ?  What  does  it  show  ?  What  are  like  signs  ? 
What  unlike  ? 


Arts.  8-17.]  INTRODUCTION.  17 

tain  cases,  comprehended  in  a  general  solution,  b  is  to  be  ad- 
ded to  a,  and  in  other  cases  subtracted  from  it. 

Obser.  When  all  the  signs  are  plus,  or  all  minus,  they  are 
said  to  be  alike ;  when  some  are  plus  and  others  minus,  they 
are  called  unlike. 

14.  The  equality  of  two  quantities,  or  sets  of  quantities, 
is  expressed  by  two  parallel  lines  =.     Thus  a-\-b—d,  signi- 
fies that  a  and  b  together  are  equal  to  d.     So  8+4rz:16-~4 
—  10+2=7+2+3. 

15.  When  the  first  of  the  two  quantities  compared,  is  great' 
er  than  the  other,  the  character  ^>  is  placed  between  them. 
Thus  d^>b  signifies  that  a  is  greater  than  b. 

If  the  first  is  less  than  the  other,  the  character  <^  is  used ; 
as  a<^b ;  i.  e,  a  is  less  than  b.  In  both  cases,  the  quantity 
towards  which  the  character  opens,  is  greater  than  the  other. 

16.  A  numeral  figure  is  often  prefixed  to  a  letter.     This  is 
called  a  co-efficient.     It  shows  how  often  the  quantity  expressed 
by  the  letter  is  to  be  taken.     Thus  2b  signifies  twice  b  ;  and 
9b,  9  times  b,  or  9  multiplied  into  b, 

The  co-efficient  may  be  either  a  whole  number  or  a  frac* 
tion.  Thus  f  b  is  two  thirds  of  b.  When  the  co-efficient  is 
not  expressed,  1  is  always  to  be  understood.  Thus  a  is  the 
same  as  la ;  i,  e.  once  a. 

17.  The  co-efficient  may  be  a  letter,  as  well  as  a  figure. 
In  the  quantity  mb,  m  may  be  considered  the  co-efficient  of 
b ;  because  b  is  to  be  taken  as  many  times  as  there  are  units 
in  m.     If  m  stands  for  6,  then  mb  is  six  times  b.     In  3abc,  3 
may  be  considered  as  the  co-efficient  of  abc ;  3«  the  co-effi- 
cient of  be  ;  or  Sab,  the  co-efficient  of  c. 

QUEST. — How  is  equality  represented  ?  How  inequality  ?  What  is 
a  co-efficient  ?  What  does  it  show  ?  When  no  co-efficient  is  express- 
ed, what  is  understood  ?  Js  the  cp-efficient  always  a  whole  number  ? 
Is  it  always  a  figure  ? 

fc     2* 


18  ALGEBRA.  [Sect.  I. 

18.  A  simple  quantity  is  either  a  single  letter  or  number, 
or  several  letters  connected  together  without  the  signs  -)-  and 
— .     Thus  a,  aft,  abd  and  8ft,  are  each  of  them  simple  quan- 
tities. 

19.  A  compound  quantity  consists  of  a  number  of  simple 
quantities  connected  by  the  sign  -f-  or  — .     Thus  a-f-ft,  d  —  y, 
b—d-{-3h,  are  each  compound  quantities.     The  members  of 
which  it  is  composed  are  called  terms. 

20.  If  there  are  two  terms  in  a  compound   quantity,  it  is 
called  a  binomial.     Thus  a-{-b  and  a— ft  are  binomials.    The 
latter  is  also  called  a  residual  quantity,  because  it  expresses 
the  difference  of  two  quantities,  or  the  remainder,  after  one  is 
taken  from  the  other.     A  compound  quantity  consisting  of 
three  terms,  is  sometimes  called  a  trinomial ;  one  of  four 
terms,  a  quadrinomial,  &c. 

21.  When  the  several  members  of  a  compound  quantity 
are  to  be  subjected  to  the  same  operation,  they  must  be  con- 
nected by  a  line  ( )  called  a  vinculum,  or  by  a  parenthe- 
sis (  ).     Thus  a— ft-f-c,  or  «—(ft-j-c),  shows  that  the  sum  of 
ft  and  c  is  to  be  subtracted  from  a.     But  a— -b-\-c  signifies 
that  ft  only  is  to  be  subtracted  from  a,  while  c  is  to  be  added. 

22.  A  single  letter,  or  a  number  of  letters,  representing 
any  quantities  with  their  relations,  is  called  an  algebraic  ex- 
pression, or  formula.     Thus  a-\-b-\-3d  is  an  algebraic  ex- 
pression. 

23.  Multiplication  is  usually  denoted  by  two  oblique  lines 
crossing  each  other  thus  X«     Thus  aXft  is  a  multiplied  into 
b  :  and  6X3  is  6  times  3,  or  6  into  3.     Sometimes  a.  point  is 

QUEST. — What  is  a  simple  quantity?  A  compound  ?  If  there  are 
two  terms,  what  is  it  called  ?  Three  ?  Four  ?  When  several  terms  are 
subjected  to  the  same  operation,  how  is  this  shown  ?  What  is  an  al- 
gebraic expression,  or  formula?  In  how  many  ways  is  multiplication 
represented  ?  First  ?  Second  ?  Third  ? 


Arts.  18-26.]  INTRODUCTION.  19 

used  to  indicate  multiplication.  Thus  a  .  b  is  the  same  as 
aXb.  But  the  sign  of  multiplication  is  more  commonly 
omitted,  between  simple  quantities  ;  and  the  letters  are  con- 
nected together  in  the  form  of  a  word  or  syllable.  Thus  db 
is  the  same  as  a  .  b  or  aXb.  And  bcde  is  the  same  as  bXc 
XdXe.  When  a  compound  quantity  is  to  be  multiplied,  a 
vinculum  or  parenthesis  is  used,  as  in  the  case  of  subtraction. 
Thus  the  sum  of  a  and  b  multiplied  into  the  sum  of  c  and  d, 
is  ^+b  X  c+3,  or  (a+l)  X  (c+d).  And  (6+2)  X  5  is 
8x5,or40.  But  6+2x5  is  6+10,  or  16.  Whenthemarks 
of  parenthesis  are  used,  the  sign  of  multiplication  is  frequent- 
ly omitted.  Thus  (x+y)  (x—y)  is  (x+y)  X  (*—#)• 

24.  When  two  or  more  quantities  are  multiplied  together, 
each  of  them  is  called  a  factor.     In  the  product  aJ,  a  is  a 
factor,  and  so  is  b.     In  the   product  a?X(a+wO,  x  is  one  of 
the  factors,  and  a-\-m  the  other.     Hence  every  co-efficient 
may  be  considered  a  factor.     (Art.  17.)     In  the  product  3y, 
3  is  a  factor  as  well  as  y. 

25.  A  quantity  is  said  to  be  resolved  into  factors,  when  any 
factors  are  taken,  which,  being  multiplied  together,  will  pro- 
duce  the   given  quantity.     Thus  3ab  may  be  resolved  into 
the  two  factors  3a  and  &,  because  3aXb  is  3#Z>.     And  5amn 
may  be  resolved  into  the  three   factors  5#,  and  m,  and  n. 
And  48  may  be  resolved  into  the  two  factors  2x24,  or  3 X 16, 
or  4 X 12,  or  6X8  ;  or  into  the  three  factors  2X3X8,  or  4X 
6X2,  &c. 

26.  Division  is  expressed  in  two  ways  :  1st.  By  a  horizon- 
tal line  between  two  dots  -f-,  which  shows  that  the  quantity 
preceding  it,  is  to  be  divided  by  that  which  follows.     Thus, 
a-^-c,  is  a  divided  by  c. 

QUEST. — What  is  a  factor  ?  When  is  a  quantity  resolved  into  fac- 
tors ?  Factors  of  Zab  ?  5amn  ?  48  ?  In  how  many  ways  is  division  ex- 
pressed ?  First  ? 


20  ALGEBRA.  [Sect.  I. 

2d.  Division  is  more  commonly  expressed  in  the  form  of  a 
fraction,  putting  the  dividend  in  the  place  of  the  numerator, 

and  the  divisor  in  that  of  the  denominator.     Thus  -  is  a  di- 

o 

vided  by  b. 

27.  When  four  quantities  are  proportional,  the  proportion 
is  expressed  by  points,  in  the  same  manner  as  in  the  Rule  of 
Three  in  arithmetic.     Thus  a  :  b  :  :  c  :  d  signifies  that  a  has  to 
b,  the  same  ratio  which  c  has  to  d.     And  ab  :  cd  : :  a-\-m  : 
b-{-n,  means  that  ab  is  to  cd,  as  the  sum  of  a  and  m,  to  the 
sum  of  b  and  n. 

28.  Algebraic  quantities  are  said  to  be  alike,  when  they 
are  expressed  by  the  same  letters,  and  are  of  the  same  power : 
and  unlike,  when  the  letters  are  different,  or  when  the  same 
letter  is  raised  to  different  powers.*     Thus  ab,  Sab,  —ab, 
and  —  6ab,  are  like  quantities,  because  the  letters  are  the 
same  in  each,  although  the  signs  and  co-efficients  are  differ- 
ent.    But  3a,  3y,  3bx,  are  unlike  quantities,  because  the  let- 
ters are  unlike,  although  there  is  no  difference  in  the  signs 
and  co-efficients.     So,  x,  xx,  and  xxx,  are  unlike  quantities, 
because  they  are  different  powers  of  the  same  quantity.  (They 
are  usually  written  x,  x2,  and  x3.)     And  universally  if  any 
quantity  is  repeated  as  a  factor  a  number  of  times  in  one  in- 
stance, and  a  different  number  of  times  in  another,  the  pro- 
ducts will  be  unlike  quantities ;  thus  cc,  cccc,  and  c,  are  un- 
like quantities.     But  if  the  same   quantity  is  repeated  as  a 
factor  the  same  number  of  times  in  each  instance,  the  products 
are  like  quantities,     Thus  aaa,  aaa,  aaa,  and  aaa,  are  like 
quantities, 

Q,UEST. — Second  ?  The  most  common  ?  How  is  proportion  ex- 
pressed ?  What  are  like  quantities  ?  Unlike  f  What  kind  of  quanti- 
ties are  3aJ  and  Gab  ?  aa  and  acia  ?  na  and  aa  ?  xxx  and  xxx  ? 

*  For  the  notation  of  poiocrs  and  roots,  see  sections  VIII,  IX. 


Arts.  27-33.]  INTRODUCTION.  21 

29.  One  quantity  is  said  to  be  a  multiple  of  another,  when 
the  former  contains  the  latter  a  certain  number  of  times  with- 
out a  remainder.     Thus  10a  is  a  multiple  of  2a  ;  and  24  is 
a  multiple  of  6. 

30.  One  quantity  is  said  to  be  a  measure  of  another,  when 
the  former  is  contained  in  the  latter  any  number  of  times, 
without  a  remainder.     Thus  3b  is  a  measure  of  155  ;  and  7 
is  a  measure  of  35. 

31.  The  value  of  an  expression,  is  the  number  or  quantity 
for  which  the  expression  stands.     Thus  the  value  of  3-|-4  is 
7  ;  of  3X4  is  12  :  of  -1/- is  2- 

32.  The  RECIPROCAL  of  a  quantity,  is  the  quotient  arising 
from  dividing  A  UNIT  by  that  quantity.     The  reciprocal  of  a 

is  - ;  the  reciprocal  of  a-\-b  is  —  — ;  the  reciprocal  of  4  is  -. 

d  CL m  I    D  TC 

33.  What  is  the  algebraic  expression  for  the  following 

t    statement,  in  which  the  letters  a,  5,  c,  &c.  may  be  supposed 
to  represent  any  given  quantities  ? 

Ex.  1.  The  product  of  «,  5,  and  c,  divided  by  the  difference 
of  c  and  d,  is  equal  to  the  sum  of  b  and  c  added  to  15  times  h. 

A   (      abc  ,  . I-IKT 

c  —  d 

2.  The  product  of  the  difference  of  a  and  h  into  the  sum 
of  Z>,  c,  and  d,  is  equal  to  37  times  w,  added  to  the  quotient 
of  b  divided  by  the  sum  of  h  and  b. 

3.  The  sum  of  a  and  Z>,  is  to  the  quotient  of  b  divided  by 
c,  as  the  product  of  a  into  c,  to  12  times  h. 

4.  The  sum  of  a,  5,  and  c,  divided  by  six  times  their  pro- 
duct, is  equal  to  four  times  their  sum  diminished  by  d. 

5.  The  quotient  of  6  divided  by  the  sum  of  a  and  5,  is  equal 
to  7  times  rf,  diminished  by  the  quotient  of  5,  divided  by  36. 

QUEST. — When  is  one  quantity  a  multiple  of  another?  When  a 
measure  ?  What  is  the  value  of  an  algebraic  expression  ?  What  is 
the  reciprocal  of  a  quantity  ? 


£2  ALGEBRA.  [Sect.  I. 

y 

34.  What  will  the  following  expressions  become,  when 

words  are  substituted  for  the  signs  ? 

6.  ?±l=abc-6m+° 
h  a-\-c 

Ans.  The  sum  of  a  and  b  divided  by  A,  is  equal  to  the 
product  of  «  ,  &,  and  c  diminished  by  6  times  m,  and  increased 
by  the  quotient  of  a  divided  by  the  sum  of  a  and  c. 


8. 
9.  a- 


771 

10        fl-fr       ,  _ 

'  ' 


aw 

35.  At  the  close  of  an  algebraic  process  it  is  often  neces- 
sary to  restore  the  numbers  for  which  letters  have  been  sub- 
stituted at  the  beginning.  In  doing  this  the  sign  X  must  not 
be  omitted  between  the  numbers,  as  it  generally  is  between 
factors  expressed  by  letters.  Thus  if  a  stands  for  3,  and  b 
for  4,  the  product  ab  is  not  34,  but  3X4,  i.  e.  12.  Suppose 
b—±  ;  c=2  ;  d—6  ;  m=S  ;  and  n=W. 
the  value  of  the  following  algebraic  expressions. 

11.       +a+mn=?^+3+8X  10=9+3+80=92.  Ans. 
19 


^ 


d  Zed 

771  ___  7) 

16.  (a+c)X(«-w)H  --  •;- 
77i  —  d 

(c+b)X(m-d) 


n—d  n  —  bc 


Arts.  34-38.]  NEGATIVE  QUANTITIES.  23 

is. 


POSITIVE     AND     NEGATIVE     QUANTITIES. 

36.  A  POSITIVE  or  AFFIRMATIVE  quantity  is  one  which  is 
to  be  added,  and  has  the  sign  -j-  prefixed  to  it.     (Art.  11.) 

37.  A  NEGATIVE  quantity  is  one  which  is  required  to  be 
SUBTRACTED,  and  has  the  sign  —  prefixed  to  it. 

When  several  quantities  enter  into  a  calculation,  it  is  fre- 
quently necessary  that  some  of  them  should  be  added  to- 
gether, while  others  are  subtracted. 

If,  for  instance,  the  profits  of  trade  are  the  subject  of  cal- 
culation, and  the  gain  is  considered  positive  ;  the  loss  will 
be  negative  ;  because  the  latter  must  be  subtracted  from 
the  former,  to  determine  the  clear  profit.  If  the  sums  of  a 
book  account  are  brought  into  an  algebraic  process,  the  debt 
and  the  credit  are  distinguished  by  opposite  signs. 

38.  The  terms  positive  and  negative,  as  used  in  the  mathe- 
matics, are  merely  relative.     They  imply  that  there  is,  either 
in  the  nature  of  the  quantities,  or  in  their  circumstances,  or 
in  the  purposes  which  they  are  to  answer  in  calculation,  some 
such  opposition  as  requires  that  one  should  be  subtracted  from 
the  other.     But  this  opposition  is  not  that  of  existence  and  non- 
existence,  nor  of  one  thing  greater  than  nothing,  and  another 
less  than  nothing.     For  in  many  cases,  either  of  the   signs 
may  be,  indifferently  and  at  pleasure,  applied  to  the  very 
same  quantity  ;  that  is,  the  two  characters  may  change  pla- 
ces.    In  determining  the  progress  of  a  ship,  for  instance,  her 

QUEST.  —  What  is  a  positive  quantity?  What  sign  has  it?  What 
is  a  negative  quantity?  What  sign  has  it?  In  business  transactions, 
how  is  the  gain  considered?  Loss?  How  are  the  terms  positive  and 
negative  used  in  mathematics  ?  Imply  what  ? 


24  ALGEBRA.  [Sect.  I. 

easting  may  be  marked  -(-?  and  her  westing—;  or  the  west- 
ing may  be+,  and  the  easting  — .  All  that  is  necessary  is, 
that  the  two  signs  be  prefixed  to  the  quantities,  in  such  a  man- 
ner as  to  show,  which  are  to  be  added,  and  which  subtracted. 
In  different  processes,  they  may  be  differently  applied.  On 
one  occasion,  a  downward  motion  may  be  called  positive,  and 
on  another  occasion  negative. 

39.  In  every  algebraic  calculation,  some  one  of  the  quan- 
tities must  be  fixed  upon  to  be  considered  positive.     All  other 
quantities  which  will  increase  this,  must  be  positive  also.     But 
those  which  will  tend  to  diminish  it,  must  be  negative.     In  a 
mercantile  concern,  if  the  stock  is  supposed  to  be  positive,  the 
profits  will  be  positive  ;  for  they  increase  the  stock  ;  they  are 
to  be  added  to  it.     But  the  losses  will  be  negative  ;  for  they 
diminish  the  stock ;  they  are  to  be  subtracted  from  it. 

40.  A  negative  quantity  is  frequently  greater,  than  the  pos- 
itive one  with  which  it  is  connected.     But  how,  it  may  be 
asked,  can  the  former  be  subtracted  from  the  latter  ?     The 
greater  is  certainly  not  contained  in  the  less :  how  then  can 
it  be  taken  out  of  it  ?     The  answer  to  this  is,  that  the  greater 
may  be  supposed  first  to  exhaust  the  less,  and  then  to  leave  a 
remainder  equal  to  the  difference  between  the  two.     If  a  man 
has  in  his  possession  1000  dollars,  and  has  contracted  a  debt 
of  1500  ;  the  latter  subtracted  from  the  former,  not  only  ex- 
hausts the  whole  of  it,   but  leaves  a  balance  of  500  against 
him.     In  common  language,  he  is  500  dollars  worse  than 
nothing. 

41.  In  this  way,  it  frequently  happens,  in  the  course  of  an 
algebraic  process,  that  a  negative  quantity  is  brought  to  stand 

Q,UEST. — How  determine  which  quantities  are  positive  ?  Negative  ? 
Is  a  negative  quantity  ever  greater  than  a  positive,  with  which  it  is 
connected?  How  subtract  the  former  from  the  latter  in  such  a  case  ? 
Give  examples.  Does  a  negative  quantity  ever  stand  alone  ?  What 
denote  ? 


Arts.  39-43.]          NEGATIVE  QUANTITIES.  -  25 

alone.  It  has  the  sign  of  subtraction,  without  being  con- 
nected with  any  other  quantity,  from  which  it  is  to  be  sub- 
tracted. This  denotes  that  a  previous  subtraction  has  left  a 
remainder,  which  is  a  part  of  the  quantity  subtracted.  If 
the  latitude  of  a  ship  which  is  20  degrees  north  of  the  equator, 
is  considered  positive,  and  if  she  sails  south  25  degrees  ;  her 
motion  first  diminishes  her  latitude,  then  reduces  it  to  noth- 
ing, and  finally  gives  her  5  degrees  of  south  latitude.  The 
sign  —  prefixed  to  the  25  degrees,  is  retained  before  the  5, 
to  show  that  this  is  what  remains  of  the  southward  motion, 
after  balancing  the  20  degrees  of  north  latitude. 

42.  A  quantity  is  sometimes  said  to  be  subtracted  from  0. 
By  this  is  meant,  that  it  belongs  on  the  negative  side  of  0. 
But  a  quantity  is  said  to  be  added  to  0,  when  it  belongs  on 
the  positive  side.     Thus,  in  speaking  of  the  degrees  of  a 
thermometer,  O-f-6  means  6  degrees  above  0  ;  and  0—6,  6 
degrees  below  0.         9 

AXIOMS. 

43.  An  AXIOM  is  a  self-evident  proposition. 

1.  If  the  same  quantity  or  equal  quantities  be  added  to 
equal  quantities,  their  sums  will  be  equal. 

2.  If  the  same  quantity  or  equal  quantities  be  subtracted 
from  equal  quantities,  the  remainders  will  be  equal. 

3.  If  equal  quantities  be  multiplied  into  the  same,  or  equal 
quantities,  the  products  will  be  equal. 

4.  If  equal  quantities  be  divided  by  the  same  or  equal 
quantities,  the  quotients  will  be  equal. 

5.  If  the  same  quantity  be  both  added  to  and  subtracted 
from  another,  the  value  of  the  latter  will  not  be  altered. 

6.  If  a  quantity  be  both  multiplied  and  divided  by  another, 
the  value  of  the  former  will  not  be  altered. 

QUEST. — What  is  meant  by  subtracting  a  quantity  from  0  ?     Added 
to  0?     What  is  an  axiom  ?    Name  some. 


26  ALGEBRA.  [Sect.  II. 

7.  Quantities  which  are  respectively  equal  to  any  other 
quantity,  are  equal  to  each  other. 

8.  The  whole  of  a  quantity  is  greater  than  a  part. 

9.  The  ivhole  of  a  quantity  is  equal  to  all  its  parts. 


SECTION    II. 

ADDITION. 

ART.  44.  Ex.  1.  John  has  x  marbles  and  gains  I  marbles 
more.  How  many  marbles  has  he' in  all  ? 

In  this  example  we  wish  to  add  x  marbles  to  b  marbles. 
But  addition  in  algebra  is  denoted  by  the  sign  +.  Hence 
x-\-T)  is  the  answer :  i.  e.  John  has  the  sum  of  x  marbles  add- 
ed to  b  marbles.  4 

2.  What  is  the  sum  of  3b  dollars  added  to  the  sum  of  c 
dollars  and/  dollars? 

By  algebraic  notation,  3b-{-c-\-f  dollars  is  the  answer. 

45.  The  learner  may  be  curious  to  know  how  many  mar- 
bles there  are  in  x-{-b  marbles ;  and  how  many  dollars  in 
3J-J-C-)-/  dollars  ?     This  depends  upon  the  number  each  let- 
ter stands  for.     But  the   questions  do  not  decide  what  this 
number  is.     It  is  not  the  object,  in  adding  them,  to  ascertain 
the  specific  value  of  x  and  y,  or  of  b,  c,  and/;  but  to  find 
an  algebraic  expression,  which  will  represent  their  sum  or 
amount.     This  process  is  called  addition.     Hence 

46.  ADDITION  in  algebra  may  be  defined,  the  connecting  of 
several  quantities  ivith  their  signs  in  one  expression. 

QUEST. — How  is  addition  denoted  ?  Write  the  sum  of  a,  b,  c,  and  d. 
What  is  this  process  called  ?  Define  addition. 


Arts.  44-50.]  ADDITION.  27 

47.  Quantities  may  be  added,  by  writing  them  one  after 
another,  without  altering  their  signs. 

N.  B.  A  quantity  to  which  no  sign  is  prefixed  is  always  to  be 
considered  positive,  i.  e.  the  sign  -|-  is  understood.     (Art.  12.) 
What  is  the  sum  of  a-\-m,  and  6  —  8,  and  2h  —  3m-\-d  ? 

a+m+b-S+Zh-Zm+d.  Ans. 

48.  It  is  immaterial  in  what  order  the  terms  or  letters  are 
arranged.     If  you  add  6  and  3  and  9,  the  amount  is  the 
same,  whether  you  put  the  6,  3,  or  9  first,  viz.  18.     But  it  is 
frequently  more  convenient  and  therefore  customary  to  arrange 
the  letters  alphabetically. 

49.  It  often  happens  that  the  expression  denoting  the  sum 
or  amount,  can  be  simplified  by  reducing  several  terms  to 
one.     Thus  the  amount  2a-)-7a-|-4a,  may  be  abridged  by 
uniting  the  three  terms  in  one.     Thus  2a  added  to  7«  is  9a, 
and  4«  added  to  9a  make  13«.     Or  2a-|-7a-t-4a—  13a. 

There  are  two  cases  in  which  reductions  can  be  made. 

50.  CASE  I.  When  the  quantities  are  alike  and  the  signs 
alike,  as  -\-4b-^-5b,  or  —  4?/  —  3y,  &c.     Add  the  co-efficients, 
annex  the  common  letter  or  letters,  and  prefix  the  common  sign. 

EXAMPLES. 

1.  What  is  the  sum  of  3a,  4a,  and  6a  ? 


3xy 

xy 
2xy 

3.  7b+  xy 
8b+Zxy 
2b+2xy 
6b-{-5xy 

4.     ry+Sabh 
3ry-\-  abh 
6ry-{-4abh 
2ry-\-  abh 

-f-6«=13«.  Ans. 
5.     cdxy-\-3mg 
2cdxy-\-  mg 
5cdxy-^-7mg 
7cdxy-\-8mg 

N.  B.  The  mode  of  proceeding  is  the  same,  when  all  the 
signs  are  —  .  Thus  —  3bc  —  be  —  5Z»c  =  —  9bc. 

QUEST.  —  How  add  quantities  ?  When  no  sign  is  prefixed  to  a  quan- 
tity, what  is  understood  ?  In  what  order  are  the  terms  or  letters 
generally  arranged  ?  Why  ?  Can  expressions  denoting  the  sum,  ever 
be  simplified  ?  How  ?  Case  first  ? 


28  ALGEBRA.  [Sect.  II. 

7.  —   ax        8.  — 2ab —   my        9.  — Sack  —  Sidy 

—  3ax  —   db  —  3my  —   ach —   My 

—  2ax  —  7aJ  —  Smy  —  5ach  —  Ibdy 
51.  CASE  II.  When  the  quantities  are  alike ,  but  the  signs 

unlike,  as  -\-9b  and  — 6b  ; 

Take  the  less  co-efficient  from  the  greater ;  to  the  difference, 
annex  the  common  letter  or  letters,  and  prefix  the  common  sign. 

Suppose  a  man's  loss  $500  and  his  gain  $2000.  The  alge- 
braic notation  is  — 500+2000,  i.  e.  $500  is  to  be  subtracted 
from  his  stock,  and  $2000  added  to  it.  But  it  will  be  the  same 
in  effect,  and  the  expression  will  be  greatly  abridged,  if  we  add 
the  difference  between  $500  and  $2000,  viz.  $1500,  to  his 
stock. 

10.  What  is  the  sum  of  I6ab  and  — *7ab  ?         Ans.  9ab. 

11.  12.  13.  14.  15. 

To     +4J  5£c  2hm      —    dy+6m      3h —  dx 

Add  — Qb     —  7bc      —  9hm          4dy—  m     5h+4dx 

53.  If  several  positive,  and  several  negative  quantities  are 
to  be  reduced  to  one  term  ;  first  reduce  those  which  are  posi- 
tive, next  those  which  are  negative,  and  then  take  the  differ- 
ence of  the  co-efficients  of  the  two  terms  thus  found. 

16.  Reduce  13b+6b+b— 4b— 5b— 7b,  to  one  term. 
13J_|_6J+5=20*  ;  and  — 46— 55— 75=z— 16£. 

Then  20£— I6b=4b.  Ans. 

17.  Add  Sxy— xy+2xy— Ixy+lxy— 9xy-\-*7xy— 6xy. 

18.  Add  Sad— 6ad+ad+*Iad— 2ad+9ad— Sad— 4ad. 

19.  Add  2abm — abm-}-7abm — Sabm-^-labm. 

20.  Add  axy — 7axy-\-Saxy — axy — Saxy-\-9axy. 

Q.UEST. — How  may  like  quantities  be  reduced  when  their  signs  are 
unlike  ?  When  several  positive  and  several  negative  quantities  are  to 
be  reduced  to  one  term,  how  proceed  ? 

--P 


Arts.  51-56.]  ADDITION.  29 

54.  If  two  equal  quantities  have  contrary  signs,  they  destroy 
each  other,  and  may  be  cancelled.    Thus  -f-66 — 66=0.    And 
(3X6)— 18=0,  so  76c— 76c=0. 

55.  If  the  letters,  or  quantities  in  the  several  terms  to  be 
added,  are  UNLIKE,  they  can  only  be  placed  after  each  other, 
with  their  proper  signs.     (Art.  47.) 

21.  If  46,  and — 6y,  and  3z,  and  17A,  and  — 5d,  and  6,  be 
added  ;  their  sum  will  be  46— 6y+3z+l7h— 5d+6. 

22.  Add  aa,  aaa,  to  xx,  xxx  and  xxxx. 

Different  letters,  and  different  powers  of  the  same  letter, 
can  no  more  be  united  in  the  same  term,  than  dollars  and 
guineas  can  be  added,  so  as  to  make  a  single  sum.  Six  guin- 
eas and  four  dollars  are  neither  ten  guineas  nor  ten  dollars. 

56.  From  the  foregoing  principles  we  derive  the  following 

GENERAL     RULE     FOR     ADDITION. 

Write  down  the  quantities  to  be  added  without  altering 
their  signs,  placing  those  that  are  alike  under  each  other ; 
and  unite  such  terms  as  are  similar, 
23.  To  3bc— 6^+26— 3^    These  may  be  arranged  thus  : 
Add—  3bc,+x— 3d+bg  Y   3bc—6d+2b—3y 
And     2d+y+3x+b    J-36c— 3d  +  * 

2d       +  y+3x 


The  sum  will  be  — 7d-f26— 2y+4x-\-bg+b 

EXAMPLES     FOR     PRACTICE, 

1.  Add  ab+ 8,  to  cd—3,  and  5ab— 4m+2, 

2.  Add  z+3y— dx,  to  7— x— 8+hni. 

3.  Add  aim — 3x-\-bm,  to  y — z+7,  and  5x-— 6y-(-9. 

QUEST. — If  two  equal  quantities  have  contrary  signs,  what  is  the 
effect  ?     If  the  letters  in  the  several  terms  are  unlike,  how  are  they  ad- 
ded ?     What  then  is  the  general  rule  for  addition  f 
3* 


30  ALGEBRA.  [Sect,  II. 

4.  Add  3aro+6— Ixy— 8,  to  lOzy— 9+5a»i. 

6.  Add  7ad — h-\-8xy — ad,  to  5ad-\-h — 7xy. 

7.  Add  Sab — 2ay-\-x,  to  ab — ay-\-bx — h. 

8.  Add  2by — 3ax-\-2a,  to  3bx — by+a. 

9.  Add  ax-\-by — xy,  to  — by-\-2xy-\-5ax. 

10.  Add  4lcdf—  lOxy — 186,  to  7  xy -\-24b-\-3cdf. 

11.  Add  36z—  17xy+18a,  to  4ax— 56x+63cx. 

12.  Add  8a6 — 66c+4ed — 7xy,  to  I7mn-\-18fg — 2ax. 

13.  Add  — 42  a6c-j-10a6c?,  to  5Qabc-\-l5abd-}-5xyz, 

14.  Add  ax— y+6— df+44,  to  4df— 20+3az+75y. 

15.  Add  45«— 106+4crff,  to  826— 4c#*+10a— 46. 

16.  Add  12(a+6)+3(a+6),  to  2(a+6)— 10(a+6). 

17.  Add  xy(a+b)+3xy(a+b),  to  2xy(a+b)— 4xy(a+b). 

18.  Add  ax-\-aa,  x-\-xxx,  4aa-\-2x-\-ax,  and  2xxx. 

19.  Add  y — yy~\-^y->  2xx-\-].Qyy,  to  4xy-\-6y — Sxx. 

20.  Add  aaa-\-4aaa,  to  lOaaa — 14aaa-}-8aaa. 

21.  Add  12yyyy — lOxx,  to  2Qxx — 8yyyj/-|-2xx-|-3yyyy. 

22.  Add  4(x— y)— 13,  to  («+6)— 16(x— y)— 7(a+6). 

23.  Add  a(x+y)— 6y,  to  40(«— 6)+8a(x+y)— 36(a— 6). 

24.  Add  10axy-±-17bcd — axy,  to  6axy — 146cc?. 

25.  Add  —  x+y+6x(a— 6)— 7x,tol6y— 15x(a-6)+25x. 

27.  Add  5a6c — Qxy-\-mn,  a-\-6abc-{-14xy — Ila-\-6mn,  to 

1 5xy —  17a6c —  1 5a — dbc-\-xy — 3»m-f-a6c. 

28.  Add  a(x+y)  -  3b(x+y)  -  4a(x+y)  -  4(x+y)  -  (x+y), 


Arts.  57-59.]  SUBTRACTION.  31 


SECTION    III. 

SUBTRACTI ON. 

ART.  57.  SUBTRACTION  in  algebra  is  finding  the  difference 
of  two  quantities  or  sets  of  quantities. 

1.  Charles  has  5a  pears,  and  James  has  3a  pears.     How 
many  more  has  Charles  than  James  ?     In  this  example  we 
wish  to  take  3a  pears  from  5a  pears.     But  subtraction  in  al- 
gebra is  denoted   by  the  sign  — .     Hence  5a—  3«  pears  rep- 
resents the  answer.     But  5«  —  3a=2«  pears.  Ans. 

2.  A  gentleman  owns  a  house  valued  at  $4500  ;  but  he  is 
in  debt  $800.     How  much  is  he  worth  ? 

$4500- $800=$3700.  Ans. 

58.  Let  us  now  attend  to  the  principle  upon  which  this  ope- 
ration is  performed.     To  illustrate  this  point,  let  us  suppose 
that  you  open  a  book  account  with  your  neighbor.     When 
footed  up,  the  debtor  side,  which  is  considered  positive,  is 
$500.     The   credit  side,   which  is  considered  negative,  is 
$300.     You  balance  the  account,  and  find  he  owes  you  $500 
—  $300— $200.     Now  if  you  take  $50  from  the  positive  or 
debtor  side,  it  will  have  the  same  effect  on  the  balance,  as 
if  you  add  $50  to  the  negative  or  credit  side.     On  the  other 
hand,   if  you  take  $50  from  the  negative  or  credit  side,  it 
will  have  the  same  effect  on  the  balance,  as  if  you  add  $50 
to  the  positive  or  debtor  side. 

59.  Hence  universally,  taking  away  a  positive  quantity 
from  an  algebraic  expression  is  the  same  in  effect  as  adding 

an  equal  negative  quantity;    and  taking  away  a  negative 
quantity,  is  the  same  as  adding  an  equal  positive  one. 


QUEST. — What  is  subtraction  ?     On  what  principle  is  the  rule  found- 
ed f    How  illustrate  this  ?     What  is  the  rule  for  subtraction  ? 


32  ALGEBRA.  [Sect.  III. 

60.  Upon  this  principle  is  founded  the  following 

GENERAL  RULE  FOR  SUBTRACTION. 

I.  Change  the  signs  of  all  the  quantities  to  be  subtracted, 

1.  e.  of  the  subtrahend,  or  suppose  them  to  be  changed  from 
+  to  — ,  or  from  —  to  +. 

II.  If  the  quantities  are  ALIKE,  unite  the  terms  as  in  addi- 
tion. (Arts.  50,  51.) 

III.  If  the  quantities  are  UNLIKE,  change  the  signs  of  the 
subtrahend,  and  write  its  terms  after  the  minuend,  as  in  ad- 
dition.  (Art.  55.) 

EXAMPLES. 

1.    From  604-96  )  Change  the  signs  (        6fl+96  )    . 

{     of  the  subtra-  \  . ,  \  AttS. 

Take  3a+46 )    hend  thus ;      \  — 3a— 46  ) 

2.  From     166     3.  Uda     4.  —28     5.  —166     6.  —  Uda 
Subtr.    126  6da          —16          —126          —  6da 

7.166    8.126    9.    6da     10, —16     11.  — 12  6  12.  —  6da 
286         166         Uda  —28  —166         —Uda 

14.  +166     15.  +Uda     16.  —28     17.  —166     18.  —Uda 
—126  —  6da  +16  +126  +  6da 

19.  From  8a6,  take  6xy.     Ans.  8a6  —  6xy. 

20.  6aay          21.  IGaaxx        22.     6dd+3d—4ddd 
Ylay  2Qax  Wdc-\-2dddd+4dy. 

62.  From  these  examples,  it  will  be  seen  that  the  difference 
between  a  positive  and  a  negative  quantity,  may  be  greater 
than  either  of  the  two  quantities.  In  a  thermometer,  the  dif- 
ference between  28  degrees  above  zero,  and  16  below,  is 
44  degrees.  The  difference  between  gaining  1000  dollars  in 
trade,  and  losing  500,  is  equivalent  to  1500  dollars. 


Arts.  60-66.]  SUBTRACTION.  33 

63.  PROOF.  —  Subtraction  may  be  proved,  as  in  arithmetic, 
by  adding  the  remainder  to  the  subtrahend.  The  sum  ought 
to  be  equal  to  the  minuend,  upon  the  obvious  principle,  that 
the  difference  of  two  quantities  added  to  one  of  them,  is  equal 
to  the  other. 

23.  From    2xy  —  1  ")  f  —  xy+7  the  subtrahend. 

Sub.    —  %y-\-7  I        Proof.  <    ^xy  —  8  remainder. 


Rem.      3xy  —  8j  (_   2xy  —  1  minuend. 

24.     h+3bx  25.     hy—ah  26.  nd—  7by 

3h—9bx  5hy—6ah  5nd—  by 

27.  28.  29.  30 

3abm  —  xy          —  17+4ax  ax-\-  76 

—20—  ax 


65.  When  there  are  several  terms  alike  in  the  subtrahend, 
they  may  be  united  and  their  sum  be  used.     Thus, 

31.  From  ab  subtract  3am-\-am-{-l  'am-{-2am-\-6am. 

Ans.  ab  —  3am  —  am  —  1am  —  2am  —  6am=ab  —  19<zwi. 

32.  From  y,  subtract  a  —  a  —  a  —  a. 

33.  From  ax  —  6c+3ox+76c,  subtract  4bc  —  2ax-\-bc-{-4ax. 

34.  From  ad-\-3dc  —  &z,  subtract  Sad+lbx  —  dc-\~ad. 

66.  The  sign  —  ,  placed  before  the  marks  of  parenthesis, 
which  include  a  number  of  quantities,  requires,  that  when 
these  marks  are  removed,  the  signs  of  all  the  quantities  thus 
included  should  be  changed. 

Thus  a  —  (6  —  c-\-d)  signifies  that  the  quantities  6,  —  c,  and 
-\-d,  are  to  be  subtracted  from  a.  Remove  the  (  )  and  the 
expression  will  then  become  a  —  6-f-c  —  d. 

35.  xy+d—  (lad—  xy+d+hm)=—  7ad+2xy—  hm. 

QJJEST.  —  How  proved  ?  When  —  is  placed  before  a  (  )  which  in- 
cludes a  quantity,  if  the  (  )  is  removed,  what  must  be  done  ? 


34  ALGEBRA.  [Sect.  III. 

67.  On  the  other  hand,  when  a  number  of  quantities  are 
introduced  within  the  marks  of  parenthesis,  with  —  immedi- 
ately preceding,  the  signs  must  be  changed. 

Thus  •— m+6— - dx+3h=— (m— b+dx— 3A). 

EXAMPLES     FOR     PRACTICE. 

1  From  6ab+1xy+l8dfg,  take  3xy+4ab+8dfg. 

2.  From — 35ax — Slab — 37m;  take — 30m — 1506 — Wax. 

3.  From  9ay+19bx+22bc  ;  take  12ay+3lbc+5Qbx. 

4.  From  8^— 10a6+6rf;  take  —\2ab+ I0d+24xy. 

5.  From  7 a+Gx+df+xyz ;  take  3z— 4a— 3df—  Ylxyz. 

6.  From  I8bc — xy+22gh  ;  take  4lxy—gh-\-bc. 

7.  From  21ax-\~y-{-ac — ay ;  take  4a — bc-\-x — yz — dc. 

8.  From  2lx+40xy— I3a ;  take  42+10a6— 5bc. 

9.  From  5xy  ;  take  2ab+3Qab+ab— 4ab. 

10.  From  5ax-{-16ay  ;  take  4ax — ay '-\-3ax-\-4ay. 

11.  Froma+6;  take  —  (c+d—f+g— h— xy). 

12.  From  7aH-16zy— 7a^;  take  —  (Gab— I2xy+ad). 

^13.  Required  to  introduce  the  following  quantities  within  a 
parenthesis  with  —  immediately  preceding,  without  altering 
their  value  ;  — a-\-b — c — d-\-f-\-gh. 

14.  Also,  ab — cdx-\-df—x — y-\-ghf—bc-\-xyz. 

15.  From  4xx+6bbb  ;  take  3xx+4bbb. 

16.  From  2Qyy — %-f  \2aaa  ;  take  I5yy — 2y — I2aaa. 

17.  From  —8(a+b)+W(x+y) ;  take  2(a+b)—6(x+y). 

18.  From  4(a+b)— 16(x— y) ;  take  17(a+6)+36(x— y). 

19.  From  2a — aa-\-ba ;  take  a — 4aa — 6ba. 

QUEST. — When  a  number  of  quantities  are  introduced  within  a  (  ) 
with  —  before  it,  what  must  be  done  ? 


Arts.  67-71.]  MULTIPLICATION.  35 

20.  From  xx+3x— xxx ;  take  2x+3xx+10xxx. 

21.  From  18— 25a6+20x+3y ;  take  3x+3y— 25a6+l. 

22.  From  6(a— y)— 17(a+y);  take  3(a+y)— 7(a— y). 

23.  From  ax — xy — my — 6  ;  take  6ax — 6xy — cy-(-46 — 7df. 

24.  From  66a— 46  ;  take  20a— b— -30«—  16a— 36+5«. 


SECTION   IV. 

MULTIPLICATION. 

ART.  68.  Ex.1.  What  will  4  lemons  cost  at  x  cents  a  piece  ? 

If  1  lemon  costs  x  cts.  4  lemons  will  cost  4  times  as  much, 
i.  e.  4x  cents.  Ans. 

2.  How  much  can  a  man  earn  in  5  months  at  a  dols.  per 
month?  Reasoning  as  before,  aX5—  5a  dols.  Ans. 

Now  4x  is  equal  to  x+z+x+x  ;  and  5a=a-\-a-\-a-\-a-{-a. 

69.  This  repeated  addition  of  a  quantity  to  itself  ',  is  called 

MULTIPLICATION. 

Obs.  From  the  definition  of  Multiplication,  it  is  manifest  that 
the  product  is  a  quantity  of  the  same  kind  as  the  multiplicand. 

70.  Multiplying  by  a  ichole  number  is  taking  the  multipli- 
cand as  many  times,  as  there  are  units  in  the  multiplier. 

Multiplying  by  1,  is  taking  the  multiplicand  once,  as  a. 
Mult,  by  2,  is  taking  the  multiplicand  twice,  as  a-f-a,  &c. 

71.  Multiplying  by  a  FRACTION  is  taking  a  certain  PORTION 
of  the  multiplicand  as  many  times,  as  there  are  like  portions 
of  a  unit  in  the  multiplier. 

Mult,  by  •£,  is  taking  £  of  the  multiplicand,  once,  as  -|a. 
Mult,  by  \  ,  is  taking  |-  of  the  multiplicand,  twice,  as 


QUEST.  —  What  is  multiplication  ?  Of  what  denomination  is  the  pro- 
duct ?  What  is  it  to  multiply  by  a  whole  number  ?  By  a  fraction  ?  By 
??  By  f?  By  ^? 


36  ALGEBRA.  [Sect.  IV. 

72.  Multiplying  two  or  more  letters  together,  is  writing 
them  one  after  the  other,  either  with,  or  without  the  sign  of 
multiplication  between  them. 

Thus  b  multipled  into  c  is  bXc,  or  b .  c,  or  be.  And  x  into 
y,  into  z,  is  xXyXz,  or  x .  y  .  z,  or  more  commonly  written 
xyz.  And  am  into  xy  is  amxy.  So  «6c  into  zyz  is  abcxyz. 

73.  It  is  immaterial  as  to  the  result  in  what  order  the  letters 
are  arranged.     The  prod,  of  ba  is  the  same  as  ab.     3  times  5 
is  equal  to  5  times  3.     The  prod,  of  a,  6  and  c,  is  dbc,  or  bac, 
or  cab,  or  cba.     It  is  more  convenient  however  to  place  the 
letters  in  alphabetical  order. 

74.  When  the  letters  have  numerical  CO-EFFICIENTS,  these 
must  be  multiplied  together,  and  prefixed  to  the  product  of  the 
letters. 

1.  Multiply  3a  into  26.    Ans.  6ab.     For  if  a  into  b  is  ab9 
then  3  times  a  into  b,  is  evidently  3«6  ;  and  if,  instead  of  mul- 
tiplying by  b,  we  multiply  by  twice  b,  the  product  must  be 
twice  as  great;  that  is,  2x3a6,  or  Gab. 

2.  Mult.   12%      3.  3dh      4.  2ad      5.  7bdh      6.  Say 
Into       2rx  my         13hmg  x  Smx 

75.  If  either  of  the  factors  consists  of  figures  only,  these 
must  be  multiplied  into  the  co-efficients  and  letters  of  the  | 
other  factors. 

Thus  3ab  into  4,  is  I2ab.  And  36  into  2x,  is  72x.  And 
24  into  %,  is  24%. 

76.  If  the  multiplicand  is  a  compound  quantity,  each  of  its 
terms  must  be  multiplied  into  the  multiplier.     Thus  b-}-c-\-d 
into  a  is  ab-{-ac-\-ad.     For  the  whole  of  the  multiplicand  is 
to  be  taken  as  many  times,  as  there  are  units  in  the  multiplier. 

QUEST. — How  are  two  or  more  letters  multiplied  together  ?     In  what 
order  are  they  arranged  ?     When  letters  have  numeral  co-efficients  ?    If 
either  factor  consists  of  figures  only  ?    If  the  multiplicand  is  a  compound    | 
quantity? 


Arts.  72-78.]  MULTIPLICATION.  37 


7.  Mult.  d+2xy    8.  2h+m    9.  3A/+1     10.  2Aro+3 
Into    36  6dy  my  46 

77.  N.  B.  The  preceding  instances  must  not  be  confoun- 
ded with  those  in  which  several  factors  are  connected  by  the 
sign  X>  or  by  a  point.     In  the  latter  case,  the  multiplier  is 
to  be  written  before  the  other  factors  without  being  repeated. 
The  product  of  6  X  d  into  a,  is  abXd,  and  not  «&X#e?.     For 
bXd  is  bd,  and  this  into  a,  is  abd.     (Art.  72.)     The  expres- 
sion &XeHs  not  to  be  considered,  like  6-j-e?,  a  compound  quan- 
tity consisting  of  two  terms.     Different  terms  are  always  sep- 
arated by  +  or  —  .    (Art.  19.)     The  product  of  bXhXmXy 
into  a,  is  a  X  6  X  A  X  w  Xy,  or  abhmy.    But  b-\-h-\-m-}-y  into  a, 
is  ab-{-ah-\-am-\-ay. 

78.  If  both  the  factors  are  compound  quantities,  each  term 
in  the  multiplier  must  be  multiplied  into  each  in  the  multipli- 
cand.    Thus  (a-\-b)  into  (c-{-d)  is  ac-\-ad-\-bc-{-bd. 

For  the  units  in  the  multiplier  a-\-b  are  equal  to  the  units 
in  a,  added  to  the  units  in  6.     Therefore  the  product  produced 
^by  a,  must  be  added  to  the  product  produced  by  b. 

The  product  of  c-\-d  into  a  is  ac-\-ad.  )      ^  A  t  76  ^ 

The  product  of  c+d  into  6  is  6c+&<M      ( 

The  product  of  c-\-d  into  a-|-6  is  therefore  ac-\-ad-\-bc-\-bd. 


11.  Mult.  3z+d     12.  4fly+26     13.     «+l     14.  26+7 
Into    2a+hm         3c  +rx  3z+4  6d+l 


15.  Mult,  d+rx+h  into 

16.  Mult.  7+66+od  into  3r+4+2A. 


QUEST.  —  Does  it  make  any  difference  in  the  result  whether  the  quan 
tities  are  connected  by  the  sign  X,  or  -j-  -?  If  both  factors  are  com- 
pound quantities,  how  proceed  ? 

4 


38  ALGEBRA.  [Sect.  IV. 

79.  When  several  terms  in  the  product  are  alike,  it  will 
be  expedient  to  set  one  under  the  other,  and  then  to  unite 
them,  by  the  rules  for  the  reduction  in  addition.  Thus, 

17.  Mult,  b+a  18.  6+c+2         19.  a+  y+l 

Into    b+ct  b+c+3  36-j-2z+7 

bb+ab 
-{-ab-{-aa 


Prod.  bb-\-2ab+aa 
20.*  Mult.  3fl+<H-4  into 

21.  Mult,  b+cd+2  into 

22.  Mult.  3b+2x+h  into 

80.  It  will  be  easy  to  see  that  when  the  multiplier  and  mul- 
tiplicand consist  of  any  quantity,  repeated  as  a  factor,  this 
factor  will  be  repeated  in  the  product,  as  many  times  as  in 
the  multiplier  and  multiplicand  together.  (Art.  163,  3.) 

23.  Mult.  aXaXa.  Here  a  is  repeated  three  times  as  a  factor. 
Into   aXa.         Here  it  is  repeated  twice. 


Prod.  aXaXaXaXa.     Here  it  is  repeated  Jive  times. 

24.  What  is  the  product  of  bbbb  into  bbb  ? 

25.  What  is  the  product  of  2a?X3#X4a:  into  5#X6#  ? 

81.  But  the  numeral  co-efficients  of  several  fellow-  factors 
should  be  brought  together  by  multiplication.     Thus 


26.  2aX36  into  4«X56  is  2aX36X4aX56,  or  l2Qaabb. 

For  the  co-efficients  are  factors,  (Art.  24,)  and  it  is  imma- 

terial in  what  order  these  are  arranged.     (Art.  73.)     So  that 

QUEST.  —  When  several  terms  in  the  product  are  alike,  how  proceed  ? 
When  the  multiplier  and  multiplicand  consist  of  the  same  factor  re- 
peated, how  many  times  will  it  be  repeated  in  the  product?  What 
should  be  done  with  numeral  co-efficients? 


Arts.  79-84.]  MULTIPLICATION.  39 


27.  The  product  of  3«X46A  into  5m  X%. 

28.  The  product  of  4&X6d  into  2x+l. 

RULE     FOR     SIGNS     IN     THE     PRODUCT. 

82.  -f-  mto  ~f"  produces  -)-  ;  —  into  -|-  gives  —  ;  -J-  into 
—  gives  —  ;  and  —  into  —  gives  -f-     That  is,  if  the  signs 
of  the  factors  are  ALIKE,  the  sign  of  the  product  will  be  af- 

firmative ;  but  if  the  signs  of  the  factors  are  UNLIKE,  the 
sign  of  the  product  will  be  negative. 

83.  The  first  case,  that  of  +  into  +,  needs  no  farther 
illustration.     The  second  is  —  into  +,  that  is,  the  multipli- 
cand is  negative,  and  the  multiplier  positive.     Thus,  —  a  into 
-f-4  is  —  4a.     For  the  repetitions  of  the  multiplicand  are, 

30.  Mult.    2a—  m        31.  h—  3d+4        32.  a—  2—  7d—  x 
Into      Sh+x  2y  3b+h 


84.  In  the  two  preceding  cases,  the  positive  sign  pre- 
fixed to  the  multiplier  shows,  that  the  repetitions  of  the  mul- 
tiplicand are  to  be  added  to  the  other  quantities  with  which 
the  multiplier  is  connected.  But  in  the  two  remaining  cases, 
the  negative  sign  prefixed  to  the  multiplier,  indicates  that 
the  sum  of  the  repetitions  of  the  multiplicand  are  to  be  sub- 
tracted from  the  other  quantities.  (Art.  70,  71.) 

Obser.  This  subtraction  is  performed,  at  the  time  of  multi- 
plying, by  making  the  sign  of  the  product  opposite  to  that  of 
the  multiplicand.  Thus  -\-a  into  —  4  is  —  4«.  For  the  repe- 
titions of  the  multiplicand  are,  -\-a-{-a-\-a-}-a—-}-4a. 

QUEST.  —  Rule  for  the  signs  ?  When  the  multiplicand  is  -}->  what 
does  it  show  ?  When  —  ,  what?  When  and  how  is  the  subtraction 
performed  ? 


40  ALGEBRA.  [Sect.  IV. 

But  this  sum  is  to  be  subtracted,  from  the  other  quantities 
with  which  the  multiplier  is  connected.  It  will  then  become 
— 4a.  (Art.  59.) 

Thus  in  the  expression  b — (4Xfl,)  it  is  manifest  that  4Xa 
is  to  be  subtracted  from  6.  Now  4X«  is  4a,  that  is  -f-4«. 
But  to  subtract  this  from  b,  the  sign  +  must  be  changed 
into  — .  So  that  6 — (4X«)  is  b— 4a.  And  «X — 4  is  there- 
fore — 4a. 

Again,  suppose  the  multiplicand  is  a,  and  the  multiplier 
(6 — 4).  As  (6 — 4)  is  equal  to  2,  the  product  will  be  equal 
to  2a.  This  is  less  than  the  product  of  6  into  a.  To  obtain 
then  the  product  of  the  compound  multiplier  (6 — 4)  into  a, 
we  must  subtract  the  product  of  the  negative  part,  from  that 
of  the  positive  part. 

33.  Multiplying  a  )  ig  ^  game  M  (  Multiplying  a 
Into                 6 — 4 J  t  Into  2 

And  the  product  6a — 4«,  is  the  same  as  the  product,  2a. 

But  if  the  multiplier  had  been  (6-f-4),  the  two  products 
must  have  been  added. 

34.  Multiplying  a  )   .    ^  game  ag  (  Multiplying  a 
Into                 6+4  >  (  Into  10 

And  the  product  6a+4a,  is  the  same  as  the  product  lOa. 

N.  B.  This  shows  at  once  the  difference  between  multiply- 
ing by  a  positive  factor,  and  multiplying  by  a  negative  one. 
In  the  former  case,  the  sum  of  the  repetitions  of  the  multipli- 
cand is  to  be  added  to,  in  the  latter  subtracted  from,  the 
other  quantities,  with  which  the  multiplier  is  connected. 
(Art.  41.) 

QUEST. — What  is  the  difference  between  multiplying  by  a  positive 
factor  and  a  negative  one  ? 


Art.  85.]  MULTIPLICATION.  41 

36.  Mult,      a+b          37.  3dy+te?+2  38.  3A+3 

Into        b — x  mr — ab  ad — 6 

85.  If  two  negatives  be  multiplied  together,  the  product 
will  be  affirmative :  — 4X—  a=+4«.  In  this  case,  as  in 
the  preceding,  the  repetitions  of  the  multiplicand  are  to  be 
subtracted,  because  the  multiplier  has  the  negative  sign. 
These  repetitions,  if  the  multiplicand  is  — a,  and  the  multi- 
plier — 4,  are  — a — a — a — a— — 4a.  But  this  is  to  be  sub- 
tracted by  changing  the  sign.  It  then  becomes  -f-4a. 

Suppose  — a  is  multiplied  into  (6 — 4).  As  6 — 4=z2,  the 
product  is,  evidently,  twice  the  multiplicand,  that  is,  — 2a. 
But  if  we  multiply  — a  into  6  and  4  separately  ;  — a  into  6 
is  — 6a,  and  — a  into  4  is  — 4«.  (Art.  83.)  As  in  the  mul- 
tiplier, 4  is  to  be  subtracted  from  6  ;  so,  in  the  product,  — 4a 
must  be  subtracted  from  — 6a.  Now  — 4a  becomes  by  sub- 
traction -\-4a.  The  whole  product  then  is  — 6a-)-4a,  which 
is  equal  to  — 2a.  Or  thus, 

39.  Multiplying    —a  )  ig  ^  S^Q  &g  (  Multiplying 
Into  6 — 4  >  (  Into 


And  the  prod.  — 6«-|-4«,  is  equal  to  the  product       — 2a. 

It  is  often  considered  a  great  mystery,  that  the  product  of 
two  negatives  should  be  affirmative.  But  it  amounts  to  no- 
thing more  than  this,  that  the  subtraction  of  a  negative  quan- 
tity, is  equivalent  to  the  addition  of  an  affirmative  one,  (Art. 
57 ;)  and,  therefore,  that  the  repeated  subtraction  of  a  nega- 
tive quantity,  is  equivalent  to  a  repeated  addition  of  an  affirma- 
tive one.  Taking  off  from  a  man's  hands  a  debt  of  ten  dol- 
lars every  month,  is  adding  ten  dollars  a  month  to  the  value 
of  his  property. 

QUEST. — Explain  how  —  into  —  gives  -J-. 
4* 


42                                                      ALGEBRA.  *••••  [Sect.  IV. 

40.  Multiply  a— 4  into  36—6. 

41.  Mult.   3ad — ah — 7  into  4 — dy — hr. 

42.  Mult.  2Ay+3m— 1  into  4rf— 2z+3. 

86.  Positive  and  negative  terms  may  frequently  "balance 

each  other,  so  as  to  disappear  in  the  product.  (Art.  54.) 

43.  Mult,  a — b          44.  mm — yy           45.  aa-\-ab-\-bb 
Into    a-{-b                 mm-\-yy  a — 6 


aa — ab 
+ab—bb 

Prod,  aa     *  — bb 

87.  For  many  purposes,  it  is  sufficient  merely  to  indicate 
the  multiplication  of  compound  quantities,  without  actually 
multiplying  the  several  terms.    Thus  (Art.  23,)  the  product  of 
a+b+c  into  h+m+y,  is  (a+b+c)X(h+m+y). 

47.  What  is  the  product  of  a-\-m  into  A-j-x  and  d-\-y  ? 
By  this  method  of  representing  multiplication,  an  important 

advantage  is  often  gained,  in  preserving  the  factors  distinct 
from  each  other. 

When  the  several  terms  are  multiplied  in  form,  the  expres- 
sion is  said  to  be  expanded. 

48.  What  does  (a-\-b)  X  (c-\-d)  become  when  expanded  ? 
89.  With  a  given  multiplicand,  the  less  the  multiplier,  the 

less  will  be  the  product.  If  then  the  multiplier  be  reduced 
to  nothing,  the  product  will  be  nothing.  Thus  aXOz=0. 
And  if  0  be  one  of  any  number  of  fellow-factors,  the  product 
of  the  whole  will  be  nothing. 

QUEST. — Is  it  always  necessary  actually  to  perform  the  multiplica- 
tion ?  What  advantage  is  gained  by  representing  it?  When  is  an 
expression  said  to  be  expanded?  When  you  multiply  a  quantity  by 
0,  what  is  the  product  ? 


Arts.  86-90.]  MULTIPLICATION.  43 

49.  What  is  the  product  of 

50.  And(a+b)X(c+d)X(h 

From  the  preceding  principles  we  derive  the  following 

GENERAL     RULE     FOR     MULTIPLICATION. 

90.  Multiply  the  letters  and  co-efficients  of  each  term  in 
the  multiplicand,  into  the  letters  and  co-efficients  of  each  term 
in  the  multiplier ;  and  prefix  to  each  term  of  the  product,  the 
sign  required  by  the  principle,  that  like  signs  produce  -f-, 
and  unlike  signs  — . 

EXAMPLES     FOR     PRACTICE. 

1.  Mult,  a+36— 2  into  4<z— 66— 4. 

2.  Mult.  4a& XzX2  into  3roy—  1+A. 

3.  Mult.  (7a/t— #)X4into4zX3X5Xrf. 

4.  Mult.  (Gab— M+l)X2into  (8+4z—  l)Xd. 

5.  Mult.  3^+y— 4+A  into  (d+x)X(h+y). 

6.  Mult.  6«z— (4A— d)  into  (6+l)x(A+l). 

7.  Mult.  7a#— l+hX(d— x)  into—  (r+3— 4m). 

8.  Mult.  a-\-b  into  a-|-6  into  a-\-b. 

9.  Mult.  2;-)-^  into  x — y  into  z-|-y. 

10.  Mult.  aa-\-bb  into  cc+rf«?  into  xx-\-yy. 

11.  Mult.  «6c — def-\-x — 7+y  into  a+6. 

12.  Mult,  xy— yy+10  into  flfl — 12< 

13.  Mult.  4(x+y)  into  3a,  into  66,  into  3. 

14.  Mult.  3(a-f-&+c^-d)  into  xyz,  into  m. 

15.  Mult.  «— 6— c+rfinto5X(c+^). 

16.  Mult.  xz-}-xy-\-yy  into  x — y. 

QUEST. — What  is  the  general  rule  for  multiplication  ? 


44  ALGEBRA.  [Sect.  V. 

17.  Mult,  aaa — 666  into  aaa-\-bbb. 

18.  Mult,  aa — ax-\-xx  into  a-\-x. 

19.  Mult,  yyy — ayy-\-aay — aaa  into  y-\-a. 

20.  Mult.  15a-f  2066  into  3a— 466, 

21.  Mult.  3a(x+y)X4intoa-f6. 

22.  Mult.  21zy— 18«+2— 7c  into  1— x. 

23.  Mult.  2az — y  into  — (6+2)  into  zyz. 

24.  Mult.  25+6a6  into  — (x—y)  into  — 2+wi. 

25.  Mult.  aa-|-2a6+66,  into  a+6  into  a+6. 


SECTION   V. 

DIVISION. 

ART.  91.  PROB.  1.  A  man  divided  4Sx  peaches  among 
6  boys.  How  many  did  each  receive  ? 

If  6  boys  receive  48x  peaches,  it  is  manifest  1  boy  will 
receive  £  of  48x  peaches ;  but  £  of  48z:=48z-;-6;=8:e 
peaches.  Ans. 

2.  If  8  hats  cost  24a  dollars,  what  will  1  hat  cost  ? 

Reasoning  as  before,  1  hat  will  cost  %  of  24a  dollars, 
viz.  3a  dollars.  Ans. 

This  process  is  called  DIVISION.  It  consists  in  finding 
how  many  times  one  quantity  contains  another ;  and  is  the 
reverse  of  multiplication,  The  quantity  to  be  divided  is 
called  the  dividend  ;  the  given  factor,  the  divisor ;  and  that 
which  is  required,  the  quotient.  Hence, 

Q.UEST. — What  is  division  ?  Of  what,  the  reverse  ?  The  quantity 
to  be  divided,  called  ?  To  divide  by  ?  The  quantity  sought  ? 


Arts.  91-95.]  DIVISION.  45 

DIVISION  is  finding  a  quotient,  which  multiplied  into  the 
divisor  will  produce  the  dividend. 

92.  As  the  product  of  the  divisor  and  quotient  is  equal  to 
the  dividend,  the  quotient  may  be  found,  by  resolving  the 
dividend  into  two  such  factors,  that  one  of  them  shall  be  the 
divisor.     The  other  will,  of  course,  be  the  quotient. 

Suppose  abd  is  to  be  divided  by  a.  The  factors  a  and  bd 
will  produce  the  dividend.  The  first  of  these,  being  a  divi- 
sor, may  be  set  aside.  The  other  is  the  quotient.  Hence, 

When  the  divisor  is  found  as  a  factor  in  the  dividend,  the 
division  is  performed  by  cancelling  this  factor. 

1.  Divide  ex  by  c.    Ans.  x.  2.  Divide  dh  by  d. 

3.  Divide    drx      4.  hmy       5.  dhxy       6.  abed     7.  abxy 
By         dr  Jim  dy  b  ax 

93.  PROOF. — Multiply  the  divisor  and  quotient  together, 
and  the  product  will  be  equal  to  the  dividend,  if  the  work  is 
right. 

Thus  ax-^-a  the  quotient  is  x.    Proof.   xXa=ax,  dividend. 

94.  If  a  letter  is  repeated  in  the  dividend,  care  must  be 
taken  that  the  factor  rejected  be  only  equal  to  the  divisor. 

8.  Divide  adb  by  a.  Ans.  ab.         9.  Divide  bbx  by  6. 

10.  Div.    aadddx      11.  aammyy      12.  aaaxxxh     13.  yyy 
By      ad  amy  aaxx  yy 

In  such  instances,  it  is  obvious  that  we  are  not  to  reject 
every  letter  in  the  dividend  which  is  the  same  with  one  in  the 
divisor. 

95.  If  the  dividend  consists  of  any  factors  whatever,  ex- 
I  punging  one  of  them  is  dividing  by  it. 

QUJCST. — When  the  divisor  is  a  factor  of  the  dividend,  how  proceed  ? 
Proof?  If  the  letter  is  repeated  in  the  dividend,  what  is  necessary  ? 
If  the  dividend  consist  of  any  factors,  what  effect  has  expunging  one 
of  them  ? 


46  ALGEBRA.  [Sect.  V, 

14.  Divide  «(6-f-d)  by  a.    Ans.  b+d. 

15.  Div.  a(b+d)    16.  (b+x)(c+d)    17.  (b+y)X(d— A)z 
By     b+d  b+x  d—h 

96.  If  there  are  numeral  co-efficients  prefixed  to  the  letters, 
the  co-efficients  of  the  dividend  must  be  divided  by  the  co- 
efficients of  the  divisor. 

18.  Divide  Gab  by  26.  Ans.  3a.      19.  Div.  16<%  by  4dx. 

20.  Div.     25dAr       21.  I2xy        22.  34drx        23.  207«» 
By  dh  6  34  m 

97.  When  a  simple  factor  is  multiplied  into  a  compound 
one,  the  former  enters  into  every  term  of  the  latter.  (Art.  76.) 
*jThus   a  into  b-{-d,  is  a6-|-«J.      Such  a  product   is   easily 
resolved   again  into  its   original   factors.      Thus  ab-\-ad= 
aX(b+d). 

25.  ab+ac+ah=aX(1>+c+h). 

26.  What  are  the  factors  of  amh-{-amx-}-amy. 

27.  What  are  the  factors  of  4ad+8ah+I2am+4ay. 
Now  if  the  whole  quantity  be  divided  by  one  of  these  fac- 
tors, according  to  Art.  95,  the  quotient  will  be  the  other  factor. 

Thus,  (ab+ad)-±a=b+d. 
29.  (ab+ad)-±(b+d)=a.     Hence, 
If  the  divisor  is  contained  in  every  term  of  a  compound 
dividend,  it  must  cancelled  in  each. 


30. 

31. 

32. 

33. 

Div.    ab-\-ac 

bdh+bdy 

aah-\-ay 

drx-\-dhx-\-dxy 

By     a 

b 

a 

dx 

QUEST. — If  there  are  numeral  co-efficients,  how  proceed  ?  When 
the  divisor  is  contained  in  every  term  of  a  compound  dividend,  how 
proceed  ? 


Arts.  96-100.]  DIVISION.  47 

34.  35.  36.  37. 

Div.   6ab+l2ac      Wdry+lQd      12Ax+8      35dm+14dz 
By     3a  2d  4  7d 

98.  On  the  other  hand,  if  a  compound  expression  contain- 
ing any  factor  in  every  term,  be  divided  by  the  other  quan- 
tities connected  by  their  signs,  the  quotient  will  be  that  fac- 
tor.    See  the  first  part  of  the  preceding  article. 

38.  39.  40.  41. 

Div.  ab+ac+ah   amh+amx+amy    4ab+Say    ahm+ahy 
By       b+c+h         h+x+y  b+2y         m+y 

99.  In   division,    as   well  as   in   multiplication,  the   cau- 
tion must  be  observed,  not  to  confound  terms  with  factors. 
(Art.  77.) 

42.  Thus  (ab+ac)-+a=b+c.     (Art.  97.) 

43.  But    (abXac)-:ra=aabc-^-a=^abc. 

44.  Quot.of(a6+ac)-HH-c).    45-  And  abXac+(bXc). 

100.  SIGNS. — In  division,  the  same  rule  is  to  be  observed 
respecting  the  signs,  as  in  multiplication ;  that  is,  if  the  divi- 
sor and  dividend  are  both  positive,  or  both  negative,  the 
quotient  must  be  positive  :   if  one  is  positive  and  the  other 
negative,  the  quotient  must  be  negative.     (Art.  82.) 

This  is  manifest  from  the  consideration  that  the  product  of 
the  divisor  and  quotient  must  be  the  same  as  the  dividend. 

46.  If   +aX+b= +a&~) 

47.  "    —aX+b— — ab[ 

48.  «    +aX-b=-  ab> 

49.  " 


QUEST.— What  caution  as  to  terms  and  factors  ?    The  rule  for  the 
signs  ? 


48  ALGEBRA.  [Sect.  V. 

50.  51.  52.  53. 

Div.     abx          Sa—  Way          Sax— Gay          GamXdh 

By      — a  — 2o  3a  — 2a 

.,  j 
101.  If  the  letters  of  the  divisor  are  not  to  be  found  in 

the  dividend,  the  division  is  expressed  by  writing  the  divisor 
under  the  dividend,  in  the  form  of  a  vulgar  fraction. 

54.  Thus  xy-^-a—-.         55.  (d—x)-. h= 


This  is  a  method  of  denoting  division,  rather  than  an  ac- 
tual performing  of  the  operation.  But  the  purposes  of  divi- 
sion may  frequently  be  answered,  by  these  fractional  expres- 
sions. As  they  are  of  the  same  nature  with  other  vulgar 
fractions,  they  may  be  added,  subtracted,  multiplied,  &c. 

102.  If  some  of  the  letters  in  the  divisor  are  in  each  term 
of  the  dividend,  the  fractional  expression  may  be  rendered 
more  simple,  by  rejecting  equal  factors  from  the  numerator 
and  denominator. 

56.        57.  58.  59.  60. 

Div.  ab         dhx         ohm — Say         ab-\-bx         2am 

By  ac         dy          ab  by  2xy 

N.  B.  These  reductions  are  made  upon  the  principle,  that  a 
given  divisor  is  contained  in  a  given  dividend,  just  as  many 
times,  as  double  the  divisor  in  double  the  dividend ;  triple  the 
divisor  in  triple  the  dividend,  &c. 

103.  If  the  divisor  is  in  some  of  the  terms  of  the  dividend, 
but  not  in  all ;  those  which  contain  the  divisor  may  be  divi- 
ded as  in  Art.  92,  and  the  others  set  down  in  the  form  of  a 
fraction. 

QUEST. — If  the  letters  of  the  divisor  are  not  found  in  the  dividend, 
how  proceed  ?  If  some  of  the  letters  in  the  divisor  are  found  in  each 
term  of  the  dividend  ?  If  the  divisor  is  in  some  of  the  terms  of  the 
dividend,  but  not  in  all  ? 


Arts.  101-105.]  DIVISION.  49 


61.  Thus  (ab+d)+a  is  either  ,  or  **+-  or  b+-. 

a  a      a  'a 

62.                       63.                  64.  65. 

Div.  dxy+rx  —  hd       2ah-\-ad-\-x        bm-\-3y  2my+dh 

By     x                          a                           —  6  2m 

104.  The  quotient  of  any  quantity  divided  by  itself  or  its 

equal,  is  obviously  a  unit.     Thus  -=1. 

a 

67.  Div.    —  .  68.  JL.  69.  f±*=?*. 

3az  4+2  a+b—  3h 

70.  71.  72.  73. 

Div.    ax+x      3bd—3d      4axy—4a+8ad       3a6+3—  6m 
By       x  3d  4a  3 

Cor.  If  the  dividend  is  greater  than  the  divisor,  the  quotient 
must  be  greater  than  a  unit  :  But  if  the  dividend  is  less  than 
the  divisor,  the  quotient  must  be  less  than  a  unit. 

74.  Divide  25  by  5.    Ans.  5.  75.  ~=4  Ans. 

DIVISION     BY     COMPOUND     DIVISORS.* 

105.  Ex.  1.  Divide  ac+bc+ad+bd,  by  a+b. 

a+b)ac+bc+ad-\-bd(c+d 

ac-\-bc,  the  first  subtrahend. 


ad+bd 

ad-\-bd,  the  second  subtrahend. 


QUEST. — To  what  is  the  quotient  of  any  quantity  divided  by  itself, 
equal  ?     Corollary  ? 

*  The  reason  for  inserting  this  article  in  the  present  place,  may  be 
learnt  from  the  preface. 

5 


50  ALGEBRA.  [Sect.  V. 

Here  ac,  the  first  term  of  the  dividend,  is  divided  by  a,  the 
first  term  of  the  divisor,  (Art.  92,)  which  gives  c  for  the  first 
term  of  the  quotient.  Multiplying  the  whole  divisor  by  this, 
we  have  ac-{-bc  to  be  subtracted  from  the  two  first  terms  of 
the  dividend.  The  two  remaining  terms  are  then  brought 
down,  and  the  first  of  them  is  divided  by  the  first  term  of 
the  divisor  as  before.  This  gives  d  for  the  second  term  of 
the  quotient.  Then  multiplying  the  divisor  by  d,  we  have 
ad+bd  to  be  subtracted,  which  exhausts  the  whole  dividend 
without  leaving  any  remainder.  (Art.  98.) 

The  rule  is  founded  on  this  principle,  that  the  product  of 
the  divisor  into  the  several  parts  of  the  quotient,  is  equal  to 
the  dividend.  (Art.  91.) 

106.  Before  beginning  to  divide,  the  terms  should  be  so 
arranged  that  the  letter,  which  is  in  thejirst  term  of  the  divi- 
sor, shall  also  be  in  the  Jirst  term  of  the  dividend.  If  this 
letter  is  repeated  as  a  factor,  either  in  the  divisor,  or  divi- 
dend, or  in  both,  the  terms  should  be  arranged  in  the  follow- 
ing order ;  put  that  term  Jirst,  ivhich  contains  this  letter  the 
greatest  number  of  times ;  the  term  containing  it  the  next 
greatest  number  of  times,  next,  and  so  on. 

2.  Divide  2aab+bbb+2abb+aaa  by  aa+bb-\-ab. 

If  we  take  aa  for  the  Jirst  term  of  the  divisor,  the  other 
terms  must  be  arranged  according  to  the  number  of  times  a 
is  repeated  as  a  factor  in  each.  Thus, 

aa+ab+bb)aaa+2aab+2abb+bbb(a+b 
aaa-\-  aab-\-  abb 

aab+  abb+bbb 
aab-\-  abb+bbb 


QUEST. — When  the  divisor  and  dividend  are  both  compound  quan- 
tities, how  arrange  the  terms  ? 


Arts.  106,  107.]  DIVISION,  51 

N.  B.  The  strictest  attention  must  be  paid  to  the  rules  for 
the  signs  in  subtraction,  multiplication,  and  division.  (Arts. 
60,  82,  100.) 

3.  Divide  xx — 2zy+yyj  by  x — y. 

4.  Divide  aa — bb,  by  a-\-b. 

5.  Divide  bb+2bc+cc,  by  b+c. 

6.  Divide  aaa-\-xxx,  by  a-j-z. 

7.  Divide  2ax — 2aax — 3aaxy-{-6aaax-\-axy — zy,by2#-y. 

8.  Divide  a-\-b — c — ax — bx-\-cx,  by  a-\-b — c. 

9.  Divide  ac+bc+ad+bd+x,  by  a+b. 
10.  Divide  ad— qh+bd— bh+y,  by  d— h. 

107.  From  the  preceding  principles  we  derive  the  following 

GENERAL     RULE     FOR     DIVISION. 


I.  DIVISION,  in  all  cases,  may  be  expressed  by  writing  the 
divisor  under  the  dividend  in  the  form  of  a  fraction. 

II.  When  the  divisor  and  dividend  are  both  simple  quanti- 
ties, and  have  letters  or  factors" common  to  each ;  divide  the 
co-efficient  of  the  divisor  by  that  of  the  dividend,  and  cancel  the 

^  factors  in  the  dividend  which  are  equal  to  those  in  the  divisor. 

III.  When  the  divisor  is  a  simple,  and  the  dividend  a  com- 
pound quantity ;  divide  each  term  of  the  dividend  by  the  divi- 
sor as  before ;  setting  down  those  terms  which  cannot  be  divi- 
ded in  the  form  of  a  fraction. 

IV.  If  the  divisor  and  dividend  are  both  compound  quan- 
tities ;  arrange  the  terms  according  to  Art.  106. 

To  obtain  thejirst  term  in  the  quotient,  divide  thejirst  term 
of  the  dividend  by  thejirst  term  of  the  divisor.  Multiply  the 
whole  divisor  by  the  term  placed  in  the  quotient ;  subtract  the 

QUEST.— What  is  the  general  ruje  for  division? 


52  ALGEBRA.  [Sect.  V. 

product  from  the  dividend ;  and  to  the  remainder,  "bring  down 
as  many  of  the  following  terms,  as  shall  be  necessary  to  con- 
tinue  the  operation.  Divide  again  by  the  jirst  term  of  the 
divisor,  and  proceed  as  before,  till  all  the  terms  of  the  divi- 
dend are  brought  down. 

V.  SIGNS. — If  the  signs  in  the  divisor  and  dividend  are 
ALIKE,  the  quotient  will  be  -f- ;  if  UNLIKE,  the  quotient  will  be— . 

EXAMPLES     FOR     PRACTICE. 

1.  Divide  12aby+6abx—  I8bbm+24b,  by  6b. 

2.  Divide  16a—  12+8y+4— 2Qadx+m,  by  4. 

3.  Divide  (a— 2A) X  (3m+y)  X#,  by  (a— 2/0  X  (3 

4.  Divide  ahd-\-4ad-\-3ay — a,  by  hd — 4d-\-3y — 1. 

5.  Divide  ax — ry-\-ad — &my — 6-f-a,  by  — a. 

6.  Divide  amy-}-3my — mxy-\-am — d,  by  — dmy. 

7.  Divide  ard— 6a+2r— M+6,  by  2ard. 

8.  Divide  6ax — 8-)-2^+4 — 6hy,  by  4axy. 

9.  Divide  IGabcx — l2xyab-{-24abxd — 36ahgl>,  by 

10.  Divide  21aaby-\-42cdxaa-\-l4:aaa — 35aaaaZ>,  by 

11.  Divide  12abxyz — 6hdabxy-\-24:xyalm,  by  3alxy. 

12.  Divide  Sax— 365^+42— 72cz+30aa?,  by  Sx. 

13.  Divide  4(kZ>— 4(x+y)+72-\-l2(a+b)+48c,  by  —4. 

14.  Divide  abx — cdx-\-8gx-\-x,  by  ab — cd-\-8g~\-l, 

15.  Divide 24xyz— 36cd— 48abcd,  by  12^2 -I8cd -24abcd. 

16.  Divide  —  ab — ad-^-ax(a-\-b) — 42axy-\-ab,  by  — a. 

17.  Divide  6am— -lOa/i+20— 12cd+17a,  by  — 2am. 

18.  Divide  xyz-\-6x+2z — l-\-2xyz(a-{-b),  by  6xyz* 

19.  Divide  — 6ac — 12Z»c — Gab — 10 — 2aabbcc,  by  — 6abc. 

20.  Divide  18abyx+16abx— 20JJcm+24a3,  by  3^. 


Art.  108.]  FRACTIONS.  53 

21.  Divide  16*— 24+8o+84— 20a0— a,  by  —4. 

22.  Divide  (.r— ?/)X(3a+a?)X^  by  (a?— y)X(3a+x). 

23.  Divide  41dX(4— a)  X  («+#)»  by  (4— a)X41cZ. 

24.  Divide — $Qxy-\-7alx — Sahmx,  by — 4Qy-\-7ab — 3ahm. 

25.  Divide  2Q(db+l)--W(ab+l)+50(ab+l),  by  5a. 

Examples  of  Compound  Quantities. 

26.  Divide  6ax+2xy— Sab — fy-|-3tfc+c#-f-A,  by  3a-|-y. 

27.  Divide  aab—3aa+2ab—6a — 4J+22,  by  £—3. 

28.  Divide  bb+3bc+2cc,  by  6+c. 

29.  Divide  Saaa—Ub,  by  2a— £. 

30.  Divide  xxx — 3axx-{-3aax — aaa,  by  x — a. 

31.  Divide  2yyy — l9yy-\-26y — 16,  by  y — 8. 

32.  Divide  xxxxxx — 1,  by  x — 1. 

33.  Divide  4xxxx— 9xx+6x— 3,  by  2zz+3z— 1. 

NOTE. — For  examples  in  dividing  compound  quantities  in 
which  the  indices  are  used,  see  Art.  194,  Exs.  23-40,  and 
Art.  196. 


SECTION    VI. 

FRACTIONS. 

ART.  108.  FRACTIONS  in  algebra,  as  well  as  in  arithmetic, 
have  reference  to  parts  of  numbers  or  quantities.  The  term 
is  derived  from  a  Latin  word,  which  signifies  broken.  Thus 

a  .  b  .  2a  .  ,  4z  . 

-  is  £«;    -  is  %b  ;  —  is  fa ;  and  —  is  fc. 

QUEST. — What  are  fractions  ?  From  what  is  the  term  derived  ? 
The  meaning  of  it  ? 

5* 


54  ALGEBRA.  [Sect.  VI. 

109.  Expressions  in  the  form  of  fractions  occur  more  fre- 
quently in  algebra  than  in  arithmetic.     Most  instances  in  di- 
vision belong  to  this  class.     Indeed  the  numerator  of  every 
fraction  may  be  considered  as  a  dividend,  of  which  the  de- 
nominator is  a  divisor. 

110.  The  value  of  a  fraction,  is  the  quotient  of  the  nume- 
rator divided  by  the  denominator. 

f\  7 

Thus  the  value  of  -  is  3.     The  value  of  -—  is  a. 
2  b 

111.  From  this  it  is  evident,  that  whatever  changes  are 
made  in  the  terms  of  a  fraction  ;  if  the  quotient  is  not  altered, 
the  value  remains  the  same.     For  any  fraction,  therefore,  we 
may  substitute  any  other  fraction  which  will  give  the  same 
quotient. 

4     10     4ba     Sdrx      6+2 


each  of  these  instances  is  2. 

1  12.  It  is  also  evident  from  the  preceding  articles,  that  if 
the  numerator  and  denominator  be  both  multiplied,  or  loth 
divided,  by  the  same  quantity,  the  value  of  the  fraction  will 
not  be  altered. 

Thus  £=§|.,  each  term  being  multiplied  by  9  ;  and  §£= 
•&=f  ,  each  term  being  divided  by  3,  and  that  quotient  by  3 


d     bx     abx     35x      £bx     -^wjju      ~ 

again,     foo  — - = zr— — =.— — — — — -;  tor  the  quotient  in 

b       ab       ob       4o        Ij"* 


each  case  is  x. 

113.  Any  integral  quantity  may,  without  altering  its  value, 
be  thrown  into  the  form  of  a  fraction,  by  making  I. the  de- 
nominator ;  or  by  multiplying  the  quantity  into  any  proposed 

QUEST. — Are  fractions  in  arithmetic  or  algebra  the  most  frequent  ? 
When  division  is  expressed  in  the  form  of  a  fraction,  where  do  you 
place  the  divisor  ?  What  is  the  value  of  a  fraction  ?  If  the  numerator 
and  denominator  are  both  multiplied,  or  both  divided  by  the  same 
quantity,  how  is  the  value  affected  ?  How  put  an  integer  into  the  form 
of  a  fraction  ? 


Arts.  109-114.]  FRACTIONS.  55 

denominator,  and  the  product  will  le  the  numerator  of  the 
fraction  required. 

m,  a     ab     ad-\-ah     6adh     .... 

Thus  a— -:=— —  =— —-.    The  quot.  of  each  is  a. 

1      6        c?+/t        6dA 

dx+hx  2drr  +  2dr 

So  rf-)-A=—     — .     And  r+l= ! . 

x  2dr 

114.  SIGNS. — (1.)  Each  sign  in  the  numerator  and  de- 
nominator of  a  fraction,  affects  only  the  single  term  to  which 
it  is  prefixed. 

(2.)  The  dividing  line  answers  the  purpose  of  a  vinculum, 
i.  e.  it  connects  the  several  terms  of  which  the  numerator 
and  denominator  may  each  be  composed. 

The  sign  prefixed  to  it,  therefore,  affects  the  whole  fraction 
collectively.  It  shows  that  the  value  of  the  whole  fraction  is 
to  be  subjected  to  the  operation  denoted  by  this  sign. 

(3.)  Hence,  if  the  sign  before  the  dividing  line  is  changed 
from  +  to  — »  or  from  —  to  -f-j  the  value  of  the  whole  frac- 
tion is  also  changed. 

The  value  of  — -  is  a.     (Art.  110.)     But  this  will  become 
6 

negative  if  the  sign  —  is  prefixed  to  the  fraction.     Thus, 
.  ab  ab 

y+-£-=y+°>    Buty— — —y— b. 

NOTE. — There  is  frequent  occasion  to  remove  the  denomi- 
nator ;  also  to  incorporate  a  fraction  with  an  integer,  or  with 
another  fraction.  In  each  of  these  cases,  if  the  sign  —  is 
prefixed  to  the  dividing  line,  all  the  signs  of  the  numerator 

QUEST. — How  far  does  the  effect  of  each  sign  in  the  numerator  and 
denominator  extend  ?  How  far,  the  sign  prefixed  to  the  dividing  line  ? 
What  does  it  show  ?  When  this  sign  is  changed,  what  is  the  effect? 
If  the  sign  —  is  prefixed  to  the  fraction,  and  you  wish  to  remove  the 
denominator,  or  to  incorporate  the  fraction  with  an  integer,  or  with 
another  fraction,  what  must  you  do  ? 


56  ALGEBRA.  [Sect.  VI. 

must  be  changed,  as  in  Art.  66,  where  a  parenthesis,  having 
the  sign  —  before  it,  is  removed. 

Thus  b 

a  a 

(4.)  If  all  the'  signs  of  the  numerator  are  changed,  the 
value  of  the  fraction  is  changed  in  the  same  manner* 

Thus  ^=+a,  (Art.  100;)  but  ^^-a.    And^=^ 
b  bo 

i    .  — ab  +  be 
=a — c ;  but = — a-\-c. 

(5.)  Again,  if  all  the  signs  of  the  denominator  are  chang- 
ed, the  value  is  also  changed. 

ab  ab 

Thus  — -=+« ;  but  — -~ — a, 

b  — b 

115.  If  then  the  sign  prefixed  to  a  fraction,  or  all  the 
signs  of  the  numerator,  or  all  the  signs  of  the  denominator 
be  changed ;  the  value  of  the  fraction  will  be  changed,  from 
positive  to  negative,  or  from  negative  to  positive. 

116.  If  any  two  of  these  changes  are  made  at  the  same 
time,  they  will  balance  each  other,  and  the  value  of  the  frac- 
tion will  not  be  altered. 

Thus  by  changing  the  sign  of  the  numerator, 

ab  — ab 

-— =-4-a  becomes  — - — = — a. 
b  b 

But  by  changing  both  the  numerator  and  denominator,  it 

becomes  -— -|-«,  where  the  positive  value  is  restored. 

—  b 

By  changing  the  sign  before  the  fraction, 

.  ab         .                            ab 
y-\ — — =y-\-a  becomes  y T~— y — &• 

Q,UEST. — If  all  the  signs  of  the  numerator,  or  of  the  denominator, 
or  the  sign  before  the  fraction  are  changed,  what  is  the  effect  ?  What 
is  the  effect  when  any  two  of  these  changes  are  made  at  the  same  time  ? 


Arts.  115-117.]  FRACTIONS.  57 

But  by  changing  the  sign  of  the  numerator  also,  it  becomes 

y  --  -  —  where  the  quotient  —  a  is  to  be  subtracted  from  y, 

or  which  is  the  same  thing,  (Art.  59,)  -j-a  is  to  be  added, 
making  y-\-a  as  at  first. 

So6--6-      ~6-       6 
'°  2-^2-  ~^    -=2 

6-6        6        -6 


Hence  the  quotient  in  division  may  be  set  down  in  different 

ways.     Thus  (a—  c)-^6,  is  either  —  I  —  —  ,  or  -—  -. 

o       o  bo 

The  latter  method  is  the  most  common.     See  the  examples 
in  Art.  103. 


REDUCTION     OF     FRACTIONS. 

117.  A  FRACTION  may  be  reduced  to  lower  terms,  by  divi- 
ding both  the  numerator  and  denominator,  by  any  quantity 
ivliich  will  divide  them  without  a  remainder. 

According  to  Art.  112  this  will  not  alter  the  value  of  the 
fraction. 

1.  Reduce  —  to  lower  terms.  Ans.  -. 

cb  c 

2.  Reduce  — .  3.  Reduce  —. 

Sdy  *lmr 

.    ._    .  a-4-bc  _,    ,.,    ,        am-\-ay 

4.  Reduce    — ~-^ .        5.  Reduce I— £. 

(a+bc)Xm  bm-^-by 

N.  B.  If  a  letter  is  in  every  term,  both  of  the  numerator 
and  denominator,  it  may  be  cancelled,  for  this  is  dividing  by 
that  letter.  (Art.  97.)  Thus, 

QUEST. — How  reduce  a  fraction  to  lower  terms  ? 


58  .ALGEBRA.  [Sect.  VI. 


6.  Reduce  Ans..      7.  Reduce 


If  the  numerator  and  denominator  be  divided  by  the  great- 
est common  measure,  it  is  evident  that  the  fraction  will  be  re- 
duced to  the  lowest  terms.  For  the  method  of  finding  the 
greatest  common  measure,  see  Art.  195. 

118.  To  reduce  fractions  of  different  denominators  to  a 
common  denominator. 

Multiply  each  numerator  into  all  the  denominators  except 
its  own  for  a  new  numerator  ;  and  all  the  denominators  to- 
gether,  for  a  common  denominator. 

8.  Reduce  -,  and  -,  and  —  to  a  common  denominator. 

b          a         y 

aXdXy=ady  } 

cXbXy=cby      ^     the  three  numerators. 

mXbXd—mbd  J 

bXdXy=bdy  the  common  denominator. 

The  fractions  reduced  are  —  -,  and  —  -,  and  —  —  . 
bdy  bay  bay 

N.  B.  It  will  be  seen,  that  the  reduction  consists  in  multi- 
plying the  numerator  and  denominator  of  each  fraction,  into 
all  the  other  denominators.  This  does  not  alter  the  value. 
(Art.  112.) 

dr         ,2h        ,  6c  ,  , 

9.  Reduce  —  ,  and  —  ,  and  —  to  a  common  denom. 

3m  g  y 

10.  Reduce  -,  and  -,  and      .      to  a  common  denom. 

3  x          d+h 

11.  Reduce  —  :  —  ,  and  -  to  a  common  denom. 

a  *f-o  a  —  b 

Q,UZST.—  -How  to  a  common  denominator  ?  Does  this  alter  the  value 
of  each  fraction  ?  Why  not  ? 


Arts.  118-120.]  FRACTIONS.  59 

An  integer  and  a  fraction,  are  easily  reduced  to  a  common 
denominator.     (Art.  113.) 

12.  Thus  a  and  -  are  equal  to  •=•  and  -,  or  —  and  -. 

c  c         c  c 

13.  Reduce  a,  6,  -,  -.  14.  Redttce  £,  -,  1 

my  b    d  f 

IK    T>   j        3s    y     1     **          -i*    T>  A        -L   x  c 
15.  Reduce  — ,  j-,  -.          rr     16.  Reduce  6,-,  -. 
a    56  2  y  * 

x    b  3c   1  10    T5  j        3x    6    x 

17.  Reduce  -,  -,  — ,  -.  18.  Reduce  — ,  — -,  -. 

a    z    y    3  a    4c   o 

a  5  8a   1  .__    _,    .        4a  ,w  y        c 

19.  Reduce  -,  -,  — ,  -.         20.  Reduce  -«-,  17,-,  x,  — -. 
o   7    y    2  Jg          c         4a 

119.  To  reduce  an  improper  fraction  to  a  whole  or  mixed 
quantity. 

Divide  the  numerator  by  the  denominator,  as  in  Art.  103. 

21.  Thus  <A+6m+d=a+m+i 
b  b 


oo    T>  j  ~- — 7tr  .      , 

22.  Reduce ,  to  a  mixed  quantity .. 

a 

120.  To  reduce  a  mixed  quantity  to  an  improper  fraction. 

Multiply  the  integer  by  the  given  denominator,  and  add  the 
given  numerator  to  the  product.  (Art.  113.)  The  sum  will 
be  the  required  numerator ;  and  this  placed  over  the  given 
denominator  will  form  the  improper  fraction  required. 

N.  B.  If  the  sign  before  the  dividing  line  is  — ,  all  the  signs 
4  in  the  numerator  must  be  changed.  (Art.  114,  note.) 

QUEST. — How  reduce  an  improper  fraction  to  a  whole  or  mixed 
quantity  ?  How  reduce  a  mixed  quantity  to  an  improper  fraction  ? 
When  the  sign  —  is  before  the  dividing  line,  what  must  be  done  ? 


60  ALGEBRA.  [Sect.  VI. 

23.  Reduce  a-f~T  to  an  improper  fraction.     Ans.      7"- . 

24.  Reduce  a .  25.  Reduce  x — 

c 

26.  Reduce  db-^—^-.  27.  Reduce 

x 

28.  Reduce  m+d — .       <*      29.  Reduce  6—  _ 

h— d  d—y 

121.  To  reduce  a  compound  fraction  to  a  simple  one. 
Multiply  all  the  numerators  together  for  a  new  numerator, 
and  all  the  denominators  for  a  new  denominator. 

30.  Reduce  ^of-?-.  Ans.  .^-rrr- 


31.  Reduce  |of^of  -          .         32.  Reduce     of    of 

3      5     2a  —  m  7      3      b-a 


EXAMPLES     FOR     PRACTICE. 

1.  What  is  the  value  of 

2.  What  is  the  value  of 


abcdf 

3.  What  is  the  value  of  —  X4  ? 

a 

Ifi 

4.  What  is  the  value  of  --  -  ~4x  ? 

a 

5.  What  is  the  value  of  —  —  when  the  denom.  is  X4  ? 

6.  What  is  the  value  of  -  —  —  when  the  denom.  is  -:-6ax  ? 

24ax 

QUEST.  —  How  reduce  a  compound  fraction  to  a  simple  one  ? 


Arts.  121,  122.]  FRACTIONS.  61 

7.  What  is  the  value  of  —  —5  when  both  numerator  and 

34a 

denominator  are  X2<#? 

8.  Reduce  -  °—^-  -  to  a  whole  or  mixed  number. 

2ab 

9.  Reduce  --  ~-  —   —  to  a  whole  mixed  or  number. 

i^x 

ab-{-e-\-dx-\-ax-\-  a  m 

10.  Reduce  —  !  —  •  -  j  -  -  -  to  a  whole  or  mixed  No. 
a 

Reduce  the  four  next  examples  to  the  lowest  terms. 


11.         .       12.  _       13. 


. 
aac  l%zyy  06+6  x  ac-\-abc 

15.  Reduce  —  and  —  to  a  common  denominator. 

y        d 

16.  Reduce  -  ;  ^  ;  -  and  -  to  a  common  denominator. 

b'  d    g          y 

17.  Reduce  a  --    -  to  an  improper  fraction. 

x 


18.  Reduce  a+6  —  -j-     to  an  improper  fraction. 

4»i 

19.  Reduce  -  of  -  of  -  of  -  to  a  simple  fraction. 

3       o       a      y 


20.  Reduce  -  of  -—  of  — -  of  —  of  ——  of  — -  to  a  sim- 
ple fraction. 

ADDITION     OF     FRACTIONS. 

122.  RULE. — Reduce  the  fractions  to  a  common  denomina- 
tor ;  then  add  their  numerators,  and  place  the  sum  over  the 
common  denominator. 

QUEST. — How  are  fractions  added  ? 

6 


62  ALGEBRA.  [Sect.  VI. 


EXAMPLES. 

1.  Add  —  and  —  of  a  pound.  Ans.  -X-  or  —  . 

J.O  1O  J.O  1O 

2.  Add  -  and  -.    First  reduce  them  to  a  common  denomi- 

o         d 

nator.    They  will  then  be  ^-7  and  ~4,  and  their  sum  —  r^—  • 
bd         ba  oa 


3.  Given  -  and  --     —  »  to  find  their  sum. 

a  3n 

a             b  —  m  ~.        a       ,     d 

4.  Given  -and  --  .  5.  Given  -  and  -  . 

ay  y          —  m 

•a          ,      b  »v     A  jj  —  a   *     —  ^ 

6.  Given  —  —  and  -  -.  7.  Add  —  —  ,  to 


:     =•     U.1J.U.      —  —•  •     *.*.****.  j     -•. 

a-\-b  a  —  6  d          m  —  r 


.   b         .   d  ,  a  —  b 

11.  Add  a+2*>    c+^;  ^  and  "4"* 

tn      A  JJ    An        %  ^+C  J  I     H~c 

12.  Add  42  --  ;  a  --  ^  —  and    a  -4-  —  {  — 


.  Add        -5=.         ; 

2c          c       2c      xy  4c 


14.  Add  2ax  ;  —   -  —  and 


123.  For  many  purposes,  it  is  sufficient  to  add  fractions  in 
the  same  manner  as  integers  are  added,  by  writing  them  one 
after  another  with  their  signs.  (Art.  47.) 

QUEST. — What  other  way  ? 


Arts.  123-125.]  FRACTIONS.  63 

15.  Thus  the  sum  of  -  and  -  and .is  — 1 . 

b         y  2m       b  r y      2m 

124.  To  add  fractions  and  integers. 

Write  them  one  after  another  with  their  signs  ;  or,  con- 
vert the  integer  into  a  fraction,  (Art.  113,)  and  then  add 
their  numerators. 

16.  What  is  the  sum  of  a,  and  -  ? 

m 

17.  What  is  the  sum  of  3d  and 


m — y 

18.  What  is  the  sum  of  5x  and 

c 

SUBTRACTION  OF  FRACTIONS. 

125.  RULE. — Change  the  signs  of  the  fractions  to  "be  sub- 
tracted from  -j-  to  — ,  and  from  —  to  -j-  ;  and  then  pro- 
ceed as  in  addition  of  fractions.  (Art.  122.) 

1.  From  -subtract  -. 
b  m 

First,  Reduce  the  fractions  to  a  common  denominator. 

C  aXm=am,  the  numerator  of  the  minuend. 
Thus,  •?  hXb  =bh,  the  numerator  of  the  subtrahend. 

(  bXm—bm,  the  common  denominator. 

The  fractions  become  —   and  - — 
bm  bm 

Second,  Change  the  sign,  before  the  dividing  line  of  the 

am       bh 

subtrahend,  as . 

bm       bm  . 

QUEST. — How  are  integers  and  fractions  added  ?  What  is  the  rule  for 
subtraction  of  fractions  ?  What  sign  do  you  change  ? 


64  ALGEBRA.  [Sect.  VI. 

Third,    Unite    the    terms    as    in  addition    of  fractions  ; 
thus, 


3.  From  -  subtract 


m  y 

,    „         a+3d 

4.  From  —  ^-  —  subtract 

4  o 

5.  From  -  subtract  —  -  . 

m  y 

6.  From     '    •  subtract  -  . 

d  m 

3  4 

7.  From  -  subtract  -. 

a  b 

126.  Fractions  may  also  be  subtracted,  like  integers,  by 
setting  them  down,  after  their  signs  are  changed,  without 
reducing  them  to  a  common  denominator. 

8.  From  *    subtract-^.     Ans.  ^ 

m  y  my 

127.  To  subtract  an  integer  from  a  fraction,  or  a  fraction 
from  an  integer. 

Change  the  sign  of  the  subtrahend,  and  write  it  after  the 
minuend  ;  or,  throw  the  integer  into  the  form  of  a  fraction, 
(Art.  113,)  and  then  proceed  according  to  the  general  ride 
for  subtraction  of  fractions. 

10.  From  -  subtract  m.    Ans.  --  m== 

y  y          y 

QUEST.  —  How  subtract  an  integer  from  a  fraction,  or  a  fraction  from 
an  integer  ? 


Arts.  126-130.]  FRACTIONS. 

11.  From  4a+-  subtract  3a—  -. 
c  a 


12.  From  1+  subtract 

a  a 

13.  From  a+3A—  d-^-  subtract  3«—  A+ 


o 

14.  From      =     take  15.  From  £±*  take 

6                  c  x                y 

16.  From  —  ^—  take  —  £—  .  17.  From  a  --  take  —  -. 

b—x          d+y  y           * 

18.  From  x+y  take  —  .  19.  From  ^  take 

c  10 


20.  From,-  take 


MULTIPLICATION     OF     FRACTIONS.     . 

128.  By  the  definition  of  multiplication,  multiplying  by  a 
fraction  is  taking  a  ^ar£  of  the  multiplicand  as  many  times 
as  there  are  like  parts  of  an  unit  in  the  multiplier.  (Art.  71.) 

o 

Suppose  a  is  to  be  multiplied  by  -. 

A  fourth  part  of  a  is  -. 

a  .  a     a     3a 

This  taken  3  times  is  7+7+7— ^T- 

444      4 

130.  Hence  ;  to  multiply  a  fraction  by  a  fraction. 
Multiply  the  numerators  together,  for  a  new  numerator, 
and  the  denominators  together,  for  a  new  denominator. 

QUEST. — What  is  meant  by  multiplying  by  a  fraction  ?  Rule  to  mul- 
tiply a  fraction  by  a  fraction  ? 

6* 


66  ALGEBRA.  [Sect.  VI. 

1.  Multiply  —  into  —  .     Product  —  . 
c  2m  2cm 


2.  Multiply  into 


y  m — 2 


o    -MT  i<-  i         --      Xh  .  .          4 
3.  Multiply  .  .  mto  -__. 


4.  Multiply  into 


5.  Multiply—  L-  into  | 

6.  Multiply  together  ^,  -   and  —  . 

DC?  y 

,,,,.,     2«    7t—  df    6,1 

7.  Multiply  —  ,   -  ,   -    and  -  . 

m        y       c  r  —  1 


8.  Multiply  ^-LZ,  -  and  -f- 


9.  Multiply  —  ,  —  —  and  -. 

hy  a-\-  1          7 

131.  The  multiplication  may  sometimes  be  shortened,  by 
rejecting  equal  factors,  from  the  numerators  and  denomin- 
ators. 

10.  Multiply  -  into  -  and  -.     Product  —  . 

ray  ry 

Here  a,  being  in  one  of  the  numerators,  and  in  one  of  the 
denominators,  may  be  omitted.  If  it  be  retained,  the  product 

will  be  --     But  this  reduced  to  lower  terms,  by  Art.  117, 
ary 

will  become  —  as  before. 


Q,UEST.  —  How  shorten  the  process,  when  the  numerators  and  deno- 
minators contain  equal  factors  ? 


Arts.  131-133.]  FRACTIONS.  67 

11     TI/T  I.L-  t     ad  •  ,  m        ,  ah 

11.  Multiply  --  into  —  -  and  —  -  . 

m  3a  %d 


12.  Multiply  into        . 

y  ah 

13.  Multiply  am  +  d  into  A  and  ?T. 

A  w  5a 

• 

132.  To  multiply  a  fraction  and  an  integer  together. 
Either  multiply  the  numerator  of  the  fraction  by  the  inte- 

ger ;  or,  divide  the  denominator  by  the  integer. 

14.  Thus  ^-Xais  -.     For  a=-  ;  and  ^X-=-.  Ans. 

y         y  l    y     y 

15.  Multiply  —  into  a.    Dividing  the  denominator  by  a,  we 

ax 

m 

have  —  . 
x 

And  multiplying  the  numerator  by  a,  we  have  —  \     But 

ax 

am      111     ,  -IP 

••  —  =.  —  ,  the  same  as  before. 
ax       x 

133.  A  fraction  is  multiplied  into  a  quantity  equal  to  its 
denominator,  by  cancelling  the  denominator.     Thus, 

_ 

16.  ^Xb=a.     For  f  Xfi—  ^-.     But  the  letter  6,  being  in 
u  ob 

both  the  numerator  and  denominator,  may  be  set  aside. 

17.  Mult.  —  into  (a—  y).     18.  Mult,  tt^  into  (3+m). 

a  —  y  o-j-m 

N.  B.  On  the  same  principle,  a  fraction  is  multiplied  into 
any  factor  in  its  denominator,  by  cancelling  that  factor. 

19.  Mult.  —  into  y.  20.  Mult.  A  into  6. 

QUEST.  —  How  multiply  a  fraction  and  an  integer  ?    How  by  a  quan- 
tity equal  to  its  denominator  ? 


68  ALGEBRA.  [Sect.  VI. 


EXAMPLES     FOR     PRACTICE. 

1.  Multiply  §,  £,  and  ™"  2.  Mult.  *  into  £±i 

*    o  *1  a         a  —  x 

3.  Mult.  ^  into  |l  4.  Mult.     3x+y     into  8. 

2  36  24a+32c 


5.  Mult.  —  Jg 
7    Mult    a^ct* 

T-  —  into  5z. 
25xy 

6.  Mult,  -into  ~  into-. 
x            o           o 

8     Mult    °           intn       ^ 

3x+y 

9.  Mult.  ~  X  -7 
6       </ 

HMult  aX 

,         ,    • 

aoca 
X  3  V  & 

/>     -     X\    "*"• 

4       a 
C  X6x 

x             a  —  6 

10.  Mult.  2+*  into  f=5. 
4               3 

1°     Mult    24fl6X3xy  y  3 

o 

x 

DIVISION     OF     FRACTIONS. 

134.  To  divide  a  fraction  by  a  fraction. 
Jnvert  the  divisor,  and  then  proceed  as  in  multiplication. 
(Art.  130.) 

1.  Divide?  by  '  Ans.  £X-  =  -. 

b       a  o       c       be 

To  understand  the  reason  of  the  rule,  let  it  be  premised, 
that  the  product  of  any  faction  into  the  same  fraction  inverted, 
is  always  a  unit. 


a      ab  hXy          d 

But  a  quantity  is  not  altered  by  multiplying  it  by  a  unit. 
Therefore,  if  a  dividend  be  multiplied,  first  into  the  divisor 
inverted,  and  then  into  the  divisor  itself,  the  last  product  will 

QUEST.  —  How  divide  one  fraction  by  another  ?  Explain  the  reason 
of  the  rule  ? 


Arts.  134-136.]  DIVISION.  69 

be  equal  to  the  dividend.  Now,  by  the  definition,  (Art.  91,) 
"  division  is  finding  a  quotient,  which  multiplied  into  the  divi- 
sor will  produce  the  dividend."  And  as  the  dividend  multi- 
plied into  the  divisor  inverted  is  such  a  quantity,  the  quotient 
is  truly  found  by  the  rule. 

_    .p..  .,     m   ,      3h  A         m  ^^  y 

2.  Divide        by      .  Ans.  -x       = 


»    TV  -j  v  i  v 

3.  Divide  —  -  —  by  —  .  4.  Divide  -  by  -  . 

r  h  x  a 


*     TV    'A  k  *     TV    'A 

5.  Divide  ——  by  -  .  6.  Divide 

5  lO 


7.  Divide  —       by 


a+I 

135.  To  divide  a  fraction  by  an  integer. 

Divide  the  numerator  by  ttie  given  integer,  when  it  can  "be 
done  without  a  remainder ;  but  when  this  can  not  be  done, 
multiply  the  denominator  by  the  integer. 

8.  Thus  the  quotient  of divided  by  m-,  is  -. 

b  b 

9.  Div.  -i-  -i_fl.     Ans ?— -.  10.  Div.  ?^6. 

a — 6  ah — oh  4 

136.  To  divide  an  integer  by  a  fraction. 

Reduce  the  integer  to  the  form  of  a  fraction,  (Art.  113,) 
and  proceed  as  in  Art.  134.  Or,  multiply  the  integer  by  the 
denominator,  and  divide  the  product  by  the  numerator.  Thus, 

QUEST. — How  many  ways  to  divide  a  fraction  by  an  integer  ?  What 
are  they  ?  How  does  it  appear,  that  multiplying  the  denominator, 
divides  a  fraction  ?  How  divide  an  integer  by  a  fraction  ? 


70  ALGEBRA.  [Sect.  VI. 


11.  Div.0-    =  =     .     Or,  «H-    =         = 

d       1       d        c  dec 

12.  Div.  x+  2±*  .  13.  Div. 


14.  Div.  3ac—  z-^- 

O 

136.a.  By  the  definition,  (Art.  32,)  "  the  reciprocal  of  a 
quantity,  is  the  quotient  arising  from  dividing  a  unit  by  that 
quantity." 

Therefore  the  reciprocal  of  £  is  1-f-  -  =  1  X  -  =  -.  That  is, 
b  b  a      a 

The  reciprocal  of  a  fraction  is  the  fraction  inverted. 

Thus  the  reciprocal  of  -  is  _JL~lL  ;    the  reciprocal   of 
m+y         b 

.  —  is  J£  or  3y  ;  the  reciprocal  of  £  is  4.     Hence  the  recip- 
&y        1 

rocal  of  a  fraction  whose  numerator  is  1,  is  the  denominator 
of  the  fraction. 

Thus  the  reciprocal  of  -  is  a  ;  of  —  —  -,  is  a+6,  &c. 
a  a-j-0 


EXAMPLES     FOR     PRACTICE. 


z—  y 

& 

• 

10—  y 

X 

QUEST. — What  is  the  reciprocal  of  a  fraction  ? 


Arts.  136.a-137.]       SIMPLE  EQUATIONS.  71 

.    .p..  .,    a+1 — x  ,      , 

4.  Divide  — — — - —  by  a. 

c    TV  -j    a-4-b ,     a 

5.  Divide  — : —  by  -. 

x  o 

x    v.,,       V 


6.  Divide  —  —  by 


7.  Divide  -—-  by 


. 
o          a  —  o 


8.  Divide  !±Jf  by  b-. 
a          a 


9.  Divide       T       by 

u 

10.  Divide  21a6c  by  —  . 

x 

11.  Divide  8xy  by  --  . 

c 

12.  Divide  18«x  by  —  (^ 

om 


.  18(a+x)  , 

13.  Divide  —  ±~  —  '-  by 

O 


14.  Divide 


SECTION   VII. 

SIMPLE     EQUATIONS 


137.  MOST  of  the  investigations  in  algebra  are  carried  on 
by  means  of  equations.  In  the  solution  of  problems,  for  ex- 
ample, we  represent  the  unknown  quantity,  or  number  sought, 


72  ALGEBRA.  [Sect.  VII. 

by  a  certain  letter ;  and  then,  to  ascertain  the  value  of  this 
unknown  quantity,  or  letter,  we  form  an  algebraic  expres- 
sion from  the  conditions  of  the  question,  which  is  equal  to 
some  given  quantity  or  number.  Thus, 

A  drover  bought  an  equal  number  of  sheep  and  cows  for 
$840.  He  paid  $2  a  head  for  the  sheep,  and  812  a  head  for 
the  cows.  How  many  did  he  buy  of  each  ? 

OPERATION. 

Let       x  —  the  number  bought  of  each. 

then   2x  =  the  price  of  all  the  sheep. 

and  I2x  =          "  "         cows.     Hence, 

2*+12ff  =  840  by  the  conditions.     (Ax.  9.) 
14#  =  840  by  uniting  the  ff's. 

and       x  =.  60,  the  number  bought  of  each. 

It  will  be  perceived  that  the  unknown  quantity  or  number 
sought,  is  represented  by  the  letter  x ;  and  from  the  conditions 
of  the  problem,  we  obtain  the  quantity  14#,  which  is  equal  to 
the  given  quantity  $840.  The  whole  algebraic  expression, 
14#=$840  is  called  an  equation. 

138.  An  equation,  then,  is  a  proposition,  expressing  in  al- 
gebraic characters,  the  equality  between  one  quantity  or  set  of 
quantities  and  another,  or  between  different  expressions  for 
the  same  quantity. 

This  equality  is  denoted  by  the  sign  — ,  which  is  read, 
"  is  equal  to,"  or  "  equals."  Thus,  x-}-a—b-\-c ;  and  5-|-8= 
17 — 4,  are  equations  in  which  the  sum  of  x  and  a  is  equal  to 
the  sum  of  b  and  c ;  and  the  sum  of  5  and  8  is  equal  to  the 

difference  of  17  and  4. 

_ 

QUEST. — How  are  investigations  generally  carried  on  in  Algebra  ? 
What  is  an  equation?  What  are  the  members  of  an  equation  ? 

"-- 


Arts.  138-141.]         SIMPLE  EQUATIONS.  73 

The  quantities  on  the  two  sides  of  the  sign  —  are  called  mem- 
bers of  the  equation  ;  the  several  terms  on  the  left,  constituting 
the  first  member,  and  those  on  the  right,  the  second  member. 

139.  When  the  unknown  quantity  is  of  the  first  power,  as 
3#,  the  proposition  is  called  a  simple  equation  ;  or  an  equation 
of  the  first  degree. 

140.  The  reduction  of  an  equation  consists,  in  "bringing  the 
unknown  quantity  by  itself,  on  one  side,  and  all  the  known 
quantities  on  the  other  side,  without  destroying  the  equality 
of  the  members. 

To  effect  this,  it  is  evident  that  one  of  the  members  must  be 
as  much  increased  or  diminished  as  the  other.     If  a  quantity 
be  added  to  one,  and  not  to  the  other,  the  equality  will  be  de- 
stroyed.    But  the  members  will  remain  equal, 
If  the  same  or  equal  quantities  be  added  to  each.     Ax.  1. 
If  the  same  or  equal  quantities  be  subtracted  from  each.  Ax.  2. 
If  each  be  multiplied  by  the  same  or  equal  quantities.     Ax.  3. 
If  each  be  divided  by  the  same  or  equal  quantities.     Ax.  4. 

140. a.  The  principal  reductions  in  simple  equations,  are 
those  which  are  effected  by  transposition,  multiplication  and 
division. 

:  REDUCTION  OF  EQUATIONS  BY  TRANSPOSITION.. 

141.  In  the   equation  x — 7=9,  the   number  7  being  con- 
nected with  the   unknown  quantity  x  by  the  sign  — ,  the  one 
is  subtracted  from   the  other.     To  reduce  the  equation,  let 

I  7  be  added  to  both  sides.     It  then  becomes  x — 7+7=9+7. 
(Art.  59.) 

The  equality  of  the  members  is  preserved,  because  one  is 
as  much  increased  as  the  other.  (Axiom  1.)  But  on  one 

QUEST. — What  is  a  simple  equation  ?  In  what  does  the  reduction 
of  an  equation  consist?  How  are  the  principal  reductions  effected? 

7 


74  ALGEBRA.  [Sect.  VII. 

side,  we  have  — 7  and  +7.     As  these  are  equal,  and  have 

contrary  signs,  they  balance  each  other,  and  may  be  cancelled. 

The  equation  will  then  be  x—9-\-7.     (Art.  54.) 

Here  the  value  of  x  is  found.     It  is  shown  to  be  equal  to 

9+7,  that  is  to   16.     The  equation  is  therefore  reduced. 

The  unknown  quantity  is  on  one  side  by  itself,  and  all  the 

known  quantities  on  the  other  side. 

In  the  same  manner,  if  x — 6=a  , 

Adding  b  to  both  sides  x — b-\-b  — a-f-6 

And  cancelling  ( — b-\-b)  x=a-\-b.     Hence, 

142.  When  known  quantities  are  connected  with  the  un- 
known quantity  by  the  sign  -f-  or  — ,  the  equation  is  reduced 
by  TRANSPOSING  the  known  quantities  to  the  other  side,  and 
and  prefixing  the  contrary  sign. 

This  is  called  reducing  an  equation  by  addition  or  subtrac- 
tion, because  it  is,  in  effect,  adding  or  subtracting  certain 
quantities,  to  or  from  each  of  the  members. 

1.  Reduce  the  equation  x-\-3b — m—h — d 
Transposing  -|-36  we  have                    x — m—h — d — 36 
And  transposing  — m,                           x—h — d — 36+w. 

143.  When  several  terms  on  the  same  side  of  an  equation 
are  alike,  they  must  be  united  in  one,  by  the  rules  for  reduc- 
tion in  addition.     (Arts.  50,  51.) 

2.  Reduce  the  equation  x+56 — 4A=76 
Transposing  56— 4A  x=^b— 56+4A 
Uniting  76 — 56  in  one  term  x=2b-}-4h. 

144.  The  unknown  quantity  must  also  be  transposed,  when- 
ever it  is  on  both  sides  of  the  equation.     It  is  not  material  on 
which  side  it  is  finally  placed. 

Q,UF.ST. — Rule  to  reduce  an  equation  by  transposition  ?  How  does 
it  appear  that  this  does  not  destroy  the  equality  of  the  members  ? 
When  several  terms  are  alike,  what  must  be  done  ?  When  the  un- 
known quantity  is  on  both  sides  of  the  equation,  what? 


Arts.  142-147.]          SIMPLE  EQUATIONS.  75 

3.  Reduce  the  equation  2z-f  2h=h-\-d-\-3x 
By  transposition  2h  —  k  —  d=3x  —  2x 
And                                                       h  —  d=  x. 

145.  When  the  same  term,  with  the  same  sign,  is  on  oppo- 
site sides  of  the  equation,  instead  of  transposing,  we  may  ex- 
punge it  from  each.     For  this  is  only  subtracting  the  same 
quantity  from  equal  quantities.     (Ax.  2.) 

4.  Reduce  the  equation  x-\-3h-\-d=b-\-3k-{-fId 
Expunging  3A  x-}-d=b-}-7d 

And  x—b+6d. 

146.  As  all  the  terms  of  an  equation  may  be  transposed, 
or  supposed  to  be  transposed,    and  it  is  immaterial  which 
member  is  written  first  ;  it  is  evident  that  the  signs  of  all  the 
terms  may  be  changed,  without  affecting  the  equality. 

Thus,  if  we  have  x  —  b=d  —  a 

Then  by  transposition  —  d-{-a—  —  x-{-b 

Or,  inverting  the  members  —  x-{-b—  —  d-\-a. 

147.  If  all  the  terms  on  one  side  of  an  equation  be  trans- 
posed, each  member  will  be  equal  to  0. 

Thus,  if  x+b=d,  then  x+6—  d=0. 

5.  Reduce  a+2x—  8=6—  4+x+a. 

6.  Reduce  y-\-ab  —  hm=a-}-2y  —  ab-}-hm. 

7.  Reduce  A+30+7z=:8—  6/i+6z—  d+b. 

S.  Reduce  M+21—  4x+d=l2—  3x+d—  76A. 
9.  Reduce  5x+10+a=25+4a;+a. 

10.  Reduce  5c+2x+  12—  3=za:+20+5c. 

11.  Reduce  a+b—3x=20+a—4x+b. 

12.  Reduce  z+3—  2z—  4z=34+3z—  4—  5x. 


QUEST.  —  When  the  same  term  with  the  same  sign  is  on  opposite 
sides,  what  ?  What  is  the  effect  when  all  the  signs  of  both  members 
are  changed  at  the  same  time?  If  all  the  terms  on  one  side  are  trans- 
posed to  the  other,  to  what  is  each  member  equal  ? 


76  ALGEBRA.  [Sect.  VII. 


REDUCTION     OF     EQUATIONS     BY     MULTIPLICATION. 

148.  The  unknown  quantity,  instead  of  being  connected 
with  a  known  quantity  by  the  sign  -|-  or  — ,  may  be  divided 

by  it,  as  in  the  equation  -  z=6. 

Here  the  reduction  can  not  be  made,  as  in  the  preceding 
instances,  by  transposition.  But  if  both  members  be  multi- 
plied by  a,  the  equation  will  become,  x=ab.  (Art.  140.) 

For  a  fraction  is  multiplied  into  its  denominator ,  by  re- 
moving the  denominator.  (Art.  133.)  Hence, 

149.  When  the  unknown  quantity  is  DIVIDED  by  a  known 
quantity,  the  equation  is  reduced  by  MULTIPLYING  every  term 
on  each  side  by  this  known  quantity. 

N.  B.  The  same  transpositions  are  to  be  made  in  this  case, 
as  in  the  preceding  examples. 

nr 

13.  Reduce  the  equation  -  -\-a 

Multiplying  both  sides  by    c 


The  product  is  x-\-ac=J)c-\-cd 

And  z=bc-\-cd — ac. 

y (I 

14.  Reduce  the  equation     - — -  +5=20. 


15.  Reduce  the  equation     — - —  -4-d=h. 
a-\-b 

150.  When  the  unknown  quantity  is  in  the  denominator  of 
a  fraction,  the  reduction  is  made  in  a  similar  manner,  by  mul- 
tiplying the  equation  by  this  denominator. 

Q,UEST. — How  is  an  equation  reduced  by  multiplication  ?  How  is  a 
fraction  multiplied  into  its  denominator  ?  How  does  it  appear  that  this 
method  of  reducing  equations  does  not  destroy  the  equality?  When 
the  unknown  quantity  is  in  the  denominator,  how  proceed  ? 


Arts.  148-153.]         SIMPLE  EQUATIONS.  77 


16.  Reduce  the  equation [-7—8. 

10 — x 

151.  Though  it  is  not  generally  necessary ,  yet  it  is  often 
convenient,  to  remove  the  denominator  from  a  fraction  con- 
sisting of  known  quantities  only.     This  may  be  done,  in  the 
same  manner,  as  the  denominator  is  removed  from  a  fraction, 
which  contains  the  unknown  quantity. 

17.  Take  for  example      -  — -+_ 

a      b        c 

TIT     i    •     i     •  i  dd     i     flh 

Multiplying  by  a        x=  —  -j 

b         c 

Multiplying  by  b        bx—ad+  — 

c 

Multiplying  by  c        bcx=acd-\-abh. 

152.  An  equation  may  be  cleared  of  FRACTIONS  by  multi- 
plying each  side  into  all  the  DENOMINATORS. 

Obser.  In  clearing  an  equation  of  fractions,  it  often  happens,  that  a 
numerator  becomes  a  multiple  of  its  denominator,  (i.  e.  can  be  divided  by 
it  without  a  remainder,)  or,  that  some  of  the  fractions  can  be  reduced 
to  lower  terms.  When  this  occurs,  the  operation  may  be  shortened  by 
performing  the  division,  and  reducing  the  fractions  to  the  lowest  terms 
according  to  Art.  117. 

18.  Reduce  the  equation     -  =  — [ , 

a       d      g        m 

19.  Reduce  the  equation     —  =  — [-  —  -ft  -, 

153.  N.  B.  In  clearing  an  equation  of  fractions,  it  will  be 
necessary  to  observe,  that  the  sign  —  prefixed  to  any  frac- 

Q.UEST. — How  clear  an  equation  of  fractions  ?  How  prove  that 
this  does  not  destroy  the  equality  ?  When  a  numerator  becomes  a 
multiple  of  its  denominator,  what  may  be  done  ?  When  a  fraction  can 
be  reduced  to  lower  terms,  what?  What  must  be  observed  as  to  the 
sign  —  before  the  dividing  line  ? 

7* 


78  ALGEBRA.  [Sect.  VII. 

tion,  denotes  that  the  whole  value  is  to  be  subtracted,  which 
is  done  by  changing  the  signs  of  all  the  terms  in  the  nume- 
rator. (Art.  114.) 

r»rt    T»    i        a — d 
20.  Reduce =c 


21.  Reduce     -  —  -r=6. 

22.  Reduce     —  r=  -  4-  —  4-  — . 

5        5  ~  5  ~  10 

23.  Reduce     2x — ~  =z  —  -\--. 

24.  Reduce     — x~l~l~l~ir —  ~^H~— -  ~ — 

REDUCTION     OF     EQUATIONS     BY     DIVISION. 

154.  When  the  unknown  quantity  is  MULTIPLIED  into  any 
known  quantity,  the  equation  is  reduced  by  DIVIDING  every 
term  on  both  sides  by  this  known  quantity. 

25.  Reduce  the  equation  ax-\-b — 3A=^ 
By  transposition  ax=.d-\-Sh — b 

TV  M-      u                                          d+3h—b 
Dividing  by  a  x= — ! . 

26.  Reduce  the  equation 

155.  If  the  unknown  quantity  has  co-efficients  in  several 
terms,  the  equation  must  be  divided  by  all  these  co-efficients, 
connected  by  their  signs,  according  to  Art.  98. 

QUEST. — What  does  this  sign,  when  thus  situated,  show  ?  When 
the  unknown  quantity  has  a  co-efficient,  how  reduce  the  equation  ? 
If  the  unknown  quantity  has  a  co-efficient  in  several  terms,  how  ? 


Arts.  154-157.]         SIMPLE  EQUATIONS.  79 

27.  Reduce  the  equation  3x — 6x=a — d 
That  is,  (Art.  97,)  (3—b)Xx=a—d. 

Dividing  by  3 — b  x—- . 

3 — b 

28.  Reduce  the  equation  ax-{-x—h — 4. 

29.  Reduce  the  equation  x — =- 

156.  If  any  quantity,  either  known  or  unknown,  is  found 
as  a  factor  in  every  term,  the  equation  may  be  divided  by  it. 
On  the  other  hand,  if  any  quantity  is  a  divisor  in  every  term, 
the  equation  may  be  multiplied  by  it.     In  this  way,  the  factor 
or  divisor  will  be  removed,  so  as  to  render  the  expression 
more  simple. 

30.  Reduce  the  equation  ax-}-3ab=:6ad-\-a 
Dividing  by  a  x-\-3b=6d-\-l 
And                                            x—6d+l—3b. 

I     1          7,         I        ..  fj 

31.  Reduce  the  equation  -21 — 

xxx 

Multiplying  by  a?,  (Art.  133,)  a;+l — b— h — d 
And  x— h— d+b— 1. 

32.  Reduce  the  equation    xX(a+b)— a— b=dX(a+b). 

157.  A  proportion  is  converted  into  an  equation  by  mak- 
ing the  product  of  the  extremes,  one  side  of  the  equation ; 
and  the  product  of  the  means,  the  other  side. 

33.  Reduce  to  an  equation  axlbl  I  chid. 
The  product  of  the  extremes  is  adx 

The  product  of  the  means  is       bch 

The  equation  is,  therefore  adx—bch. 

34.  Reduce  to  an  equation  a-\-blcl  ih — mly. 

QUEST. — If  any  quantity  is  found  as  a  factor  in  every  terra,  how  ? 
How  convert  a  proportion  into  an  equation  ? 


80  ALGEBRA.  [Sect.  VII. 

158.  On  the  other  hand,  an  equation  may  be  converted  into 
a  proportion,  by  resolving  one  side  of  the  equation  into  two 
factors,  for  the  middle  terms  of  the  proportion  ;  and  the 
other  side  into  two  factors,  for  the  extremes. 
'  35.  Convert  the  equation,  adx^^bch^  into  a  proportion. 
The  first  member  may  be  divided  into  the  two  factors  ax, 
and  d  ;  the  second  into  ch,  and  b.  From  these  factors  we 
may  form  the  proportion  ax  ib  1  1  chid. 

36.  Reduce  ay-\-by=ch  —  cm. 

37.  Reduce  16z+2=:34. 

38.  Reduce  4x—  8=—  3z+13. 

39.  Reduce  lOz—  19=7a?+17. 

40.  Reduce  Sx—  3+9=—  7*+9+27. 

SUBST  I  TUTI  ON. 

159.  In  the  reduction  of  an  equation,  as  well  as  in  other 
parts  of  algebra,  a  complicated  process  can  often  be  rendered 
shorter  and  more  simple,  by  using  letters  for  the  given  num- 
bers when  large,  (Art.  35  ;)  and  also  by  introducing  a  new 
letter  which  shall  be  made  to  represent  a  whole  algebraic  ex- 
pression. 

160.  This  process  is  called  SUBSTITUTION.     After  the  ope- 
ration is  completed,  the  numbers,  or  the  compound  quantity  for 
which  a  single  letter  has  been  substituted,  must  be  restored. 

41.  Reduce  -  -\  --  =1.     Clearing  of  fractions, 

o  7  o 


375z+3  X  750=  1  X  750  X  375 

,        281250—2250     „..     . 
and  x=i  -  —  -  -  :=744.  Ans. 
375 


QUEST.  —  How  can  an  equation  be  converted  into  a  proportion  ? 
What  is  meant  by  substitution  ?  What  is  the  advantage  of  it  ?  After 
the  operation  is  performed,  what  must  be  done  ? 


Arts.  158-160.]         SIMPLE  EQUATIONS.  81 

By  substituting  a  for  750  ;  b  for  3  ;  and  c  for  375  ;  the  equa- 

tion becomes  —  I  —  =1. 
a     c 

Clearing  of  fractions,  cz-\-ab=ac  :  and  x=a  —  —  . 

3X750 

Restoring  the  numbers,  a?=z750  --          —744.  Ans. 


42.  Reduce  —  +  6  =  84.      Substitute  a  for  3  ;    b  for  4  ; 


c  for  6  ;  and  d  for  84. 

y>  £ 

43.  Reduce  s~rH 
4500  ;  c  for  7000  ;  and  d  for  10. 


r        4500 

43.  Reduce  ^+^=10.     Substitute  a  for  350;  b  for 
ooO     7000 


44.  Reduce  -JL_-  [-^—  &.  Substitute  d  for  (ro+w),  and  the 
m-\-n     c 

.     x    ,   a      , 
equation  is  -  -f-  -  =  o. 
df       c 

Clearing  of  fractions,  ex  +  ad  =  &cd  ;  and  x  rz  — 

6c(»i-|- 
restoring  (m  +  n)  ;  x=  - 


45.  Reduce  —        —  |  —  =ra5.     Substitute  h  for  (l-m-n). 

I  —  m  —  n     c 

46.  Reduce  -  -  T  .  a  ,    —cd.    Substitute  h  for 

m     b-\-  c  -j-a 

EXAMPLES     FOR     PRACTICE. 

1.  Reduce 


2.  Reduce  -+£--- 

a  '        6     c 

3.  Reduce  40—  6x—  16=120—  14x. 


82  ALGEBRA.  [Sect.  VII. 

4.  Reduce 


5.  Reduce    +f-20—  -. 
3     5  4 


6.  Reduce          —  4=5. 

x 

7.  Reduce  -4-:  —  2=8. 

z+4 

8.  Reduce  -^U=l. 

z+4 

9.  Reduce  *=ll. 


10.  Reduce    +1-1. 

n      -D    J          *  -  5  284  -  X 

11.  Reduce—- 

4 

12.  Reduce 


13.  Reduce  -2= 

i^    -D  j 

14.  Reduce 


. 
3 

—n     5x—  5  ,  97—7* 


15.  Reduce  Jto-          -  4=  -  -. 

4  o  1>« 


16.  Reduce  - 

17.  Reduce  ^=^ 

53 

18. 


2 


* 

Art.  161.]  SIMPLE  EQUATIONS.  83 

6r.+7  .  7z— 13 
iy.  Reduce  — 


20.  Reduce^-1—:-     -::7:4. 
2  4 

SOLUTION     OF     PROBLEMS. 

161.  For  the  solution  of  problems  in  Simple  Equations,  we 
derive  from  the  preceding  principles,  the  following 

GENERAL     RULE. 

I.  Translate  the  statement  of  the  question  from  common 
to  algebraic  language,  in  such  a  manner  as  to  form  an  equa* 
tion,  i.  e.  put  the  question  into  an  equation.    (Art.  33.) 

II.  Clear  the  equation  of  fractions  by  multiplying  every 
term  on  each  side  by  all  the  denominators.    (Art.  152.) 

III.  Transpose  all  the  terms  containing  the  unknown  quan~ 
tity  to  one  side,  and  all  the  known  quantities  to  the  other, 
taking  care  to  change  the  signs  of  the  terms  transposed,  and 
unite  the  terms  that  are  alike.    (Arts.  50,  51.) 

IV.  Remove  the  co-efficients  of  the  unknown  quantity,  by 
dividing  all  the  terms  in  the  equation  by  them.    (Art.  154.) 

PROOF. — Substitute  the  value  of  the  unknown  quantity  for 
the  letter  itself  in  the  operation  ;  and  if  the  number  satisfies 
the  conditions  of  the  question,  it  is  the  answer  sought. 

Problem  1.  A  man  being  asked  how  much  he  gave  for  his 
watch,  replied ;  If  you  multiply  the  price  by  4,  and  to  the 
product  add  70,  and  from  this  sum  subtract  50,  the  remainder 
will  be  equal  to  220  dollars. 

To  solve  this,  we  must  first  translate  the  conditions  of 
the  problem,  into  such  algebraic  expressions  as  will  form  an 
equation. 

QUEST. — What  is  the  first  step  in  the  solution  of  a  problem  ?  Se- 
cond ?  Third  ?  Fourth  ?  Proof? 


84  ALGEBRA.  [Sect.  VII. 

Let  the  price  of  the  watch  be  represented  by     x 

This  price  is  to  be  mult'd  by  4,  which  makes    4x 

To  the  product,  70  is  to  be  added,  making       4z-|-70 

From  this,  50  is  to  be  subtracted,  making        4x-j-70  —  50. 

Here  we  have  a  number  of  the  conditions,  expressed  in 

algebraic  terms  ;  but  have  as  yet  no  equation.     We  must.  ob- 

serve then,  that  by  the  last  condition  of  the  problem,  the  pre- 

ceding terms  are  said  to  be  equal  to  220. 

We  have,  therefore,  this  equation     4x+70  —  50i=220 

Which  reduced  gives  s=50. 

Here  the  value  of  x  is  found  to  be  50  dollars,  which  is  the 

price  of  the  watch. 

Proof.  —  The  original  equation  is       4z+70  —  50—220 
Substituting  50  for  x,  it  becomes        4X^0+70  —  50=220 
That  is,  220=220. 

Prob.  2.  What  number  is  that,  to  which,  if  its  half  be  added, 
and  from  the  sum  20  be  subtracted,  the  remainder  will  be  a 
fourth  of  the  number  itself  ? 

In  stating  questions  of  this  kind,  where  fractions  are 
concerned,  it  should  be  recollected,  that  ^x  is  the  same  as 

|  ;    that  f  x—  ~,  &c.     (Art.  108.) 
3  5 

In  this  problem,  let  x  be  put  for  the  number  required. 

CC  T 

Then  by  the  conditions  proposed,     x-\--  —  20=- 

Z  4 

And  reducing  the  equation  or=16. 

Proof, 


Prob.  3.  A  father  divides  his  estate  among  his  three  sons, 
in  such  a  manner,  that, 

The  first  has  $  1000  less  than  half  the  whole  ; 

The  second  has  800  less  than  one  third  of  the  whole  ; 


Arts.  162,  162.0.]       SIMPLE  EQUATIONS.  85 

The  third  has  600  less  than  one  fourth  of  the  whole  ; 

What  is  the  value  of  the  estate  ? 

Prob.  4.  Divide  48  into  two  sucH  parts,  that  if  the  less  be 
divided  by  4,  and  the  greater  by  6,  the  sum  of  the  quotients 
will  be  9. 

Here,  if  x  be  put  for  the  smaller  part,  the  greater  will  be 
48—  ar. 

By  the  conditions  of  the  problem     -  -I  --  -  —  =9. 

4  o 

162.  Letters  may  be  employed  to  express  the  known  quan- 
tities in  an  equation,  as  well  as  the  unknown.  (Art.  159.) 
A  particular  value  is  assigned  to  the  numbers,  when  they  are 
introduced  into  the  calculation  ;  and  at  the  close,  the  num- 
bers are  restored.  (Art.  35.) 

Prob.  5.  If  to  a  certain  number,  720  be  added,  and  the 
sum  be  divided  by  125  ;    the  quotient  will   be  equal  to  7392 
divided  by  462.     What  is  that  number  ? 
Let  x—  the  number  required. 


Then  by  the  conditions  of  the  problem        '      —  -. 

b      ;    h 

Therefore  a=  **=?* 

A 

(125X7392)  -(720X462) 
Restoring  the  numbers,  a?=  -  --  —r-5  —  —=  1280. 

162.  a.  When  the  solution  of  an  equation  brings  out  a  nega- 
tive answer,  it  shows  that  the  value  of  the  unknown  quantity 


QUEST. — When  letters  are  substituted  for  known  quantities,  what  must 
be  done  at  the  close  of  the  calculation  ?  When  the  solution  brings  out 
a  negative  answer,  what  does  it  show  ? 

8 


86  ALGEBRA.  [Sect.  VII. 

is  contrary  to  the  quantities,  which  in  the  statement  of  the 
question  were  considered  positive.  But  this  being  deter- 
mined by  the  answer,  the  omission  of  the  sign  —  before  the 
unknown  quantity  in  the  course  of  the  calculation,  can  lead 
to  no  mistake. 

Prob.  6.  A  merchant  gains  or  loses,  in  a  bargain,  a  certain 
sum.  In  a  second  bargain,  he  gains  350  dollars,  and,  in  a 
third,  loses  60.  In  the  end  he  finds  he  has  gained  200  dol- 
lars, by  the  three  together.  How  much  did  he  gain  or  lose 
by  the  first  ? 

In  this  example,  as  the  profit  and  loss  are  opposite  in  their 
nature,  they  must  be  distinguished  by  contrary  signs.  (Art. 
39.)  If  the  profit  is  marked  -)-,  the  loss  must  be  — . 

Let  #1=  the  sum  required. 

Then  according  to  the  statement          #-(-350 — 60=200 

And  x=— 90. 

Prob.  7.  A  ship  sails  4  degrees  north,  then  13  S.  then  17 
N.  then  19  S.  and  has  finally  11  degrees  of  south  latitude. 
What  was  her  latitude  at  starting  ? 

Prob.  8,  If  a  certain  number  is  divided  by  12,  the  quotient, 
dividend,  and  divisor,  added  together,  will  amount  to  64. 
What  is  the  number  ? 

Prob.  9.  An  estate  is  divided  among  four  children,  in  such 
a  manner  that 

The  first  has  200  dollars  more  than  £  of  the  whole, 

The  second  has  340  dollars  more  than  |  of  the  whole, 

The  third  has  300  dollars  more  than  £  of  the  whole, 

The  fourth  has  400  dollars  more  than  £  of  the  whole. 

What  is  the  value  of  the  estate  ? 

Prob.  10.  What  is  that  number  which  is  as  much  less  than 
500,  as  a  fifth  part  of  it  is  greater  than  40  ? 

Prob.  11.  There  are  two  numbers  whose  difference  is  40, 
and  which  are  to  each  other  as  6  to  5.  What  are  the  numbers  ? 


Art.  162.a.]  SIMPLE  EQUATIONS.  87 


b.  12.  Three  persons,  A,  B,  and  C,  draw  prizes  in  a 
lottery.  A  draws  200  dollars  ;  B  draws  as  much  as  A,  to- 
gether with  a  third  of  what  C  draws  ;  and  C  draws  as  much 
as  A  and  B  both.  What  is  the  amount  of  the  three  prizes  ? 

Prob.  13.  What  number  is  that,  which  is  to  12  increased 
by  three  times  the  number,  as  2  to  9  ? 

Prob.  14.  A  ship  and  a  boat  are  descending  a  river  at  the 
same  time.  The  ship  passes  a  certain  fort,  when  the  boat  is 
13  miles  below.  The  ship  descends  five  miles,  while  the 
boat  descends  three.  At  what  distance  below  the  fort  will 
they  be  together  ? 

Prob.  15.  What  number  is  that,  a  sixth  part  of  which  ex- 
ceeds an  eighth  part  of  it  by  20  ?  /  •  ( 

Prob.  16.  Divide  a  prize  of  2000  dollars  into  two  such 
parts,  that  one  of  them  shall  be  to  the  other,  as  9:7.A;  )^_/,'< 

Prob.  17.  What  sum  of  money  is  that,  whose  third  part, 
fourth  part,  and  fifth  part,  added  together,  amount  to  94 
dollars.  /  -[  , 

Prob.  18.  Two  travellers,  A  and  J5,  360  miles  apart,  travel 
towards  each  other  till  they  meet.  A^s  progress  is  10  miles 
an  hour,  and  .ZTs  8.  How  far  does  each  travel  before  they 
meet  ? 

Prob.  19.  A  man  spent  one  third  of  his  life  in  England, 
one  fourth  of  it  in  Scotland,  and  the  remainder  of  it,  which 
was  20  years,  in  the  United  States.  To  what  age  did  he  live  ?  &,{ 

Prob.  20.  What  number  is  that  ^  of  which  is  greater  than 
|  of  it  by  96  ?  /  (,» 

Prob.  21.  A  post  is  ^  in  the  earth,  ^  in  the  water,  and  13 
feet  above  the  water.  What  is  the  length  of  the  post  ? 

Prob.  22.  What  number  is  that,  to  which  10  being  added, 

of  the  sum  will  be  66  ? 


88  ALGEBRA.  [Sect.  VII. 

Prob.  23.  Of  the  trees  in  an  orchard,  f  are  apple  trees,  -fa 
pear  trees,  and  the  remainder  peach  trees,  which  are  20 
more  than  £  of  the  whole.  What  is  the  whole  number  in  the 
orchard  ? 

Prob.  24.  A  gentleman  bought  several  gallons  of  wine  for 
94  dollars ;  and  after  using  7  gallons  himself,  sold  £  of  the 
remainder  for  20  dollars.  How  many  gallons  had  he  at  first  ? 

Prob.  25.  A  and  B  have  the  same  income.  A  contracts 
an  annual  debt  amounting  to  nf  of  it ;  B  lives  upon  f  of  it ; 
at  the  end  of  ten  years,  B  lends  to  A  enough  to  pay  off  his 
debts,  and  has  160  dollars  to  spare.  What  is  the  income  of 
each  ? 

M  Prob.  26.  A  gentleman  lived  single  £  of  his  whole  life  ; 
and  after  having  been  married  5  years  more  than  ~f  of  his  life, 
he  ha^lva  son  who  died  4  years  before  him,  and  who  reached 
only  half  the  age  of  his  father.  To  what  age  did  the  father 
live  ?  -^ 

Prob.  27.  What  number  is  that,  of  which  if  £,  £,  and  f  be 
added  together  the  sum  will  be  73  ? 

Prob.  28.  A  person  after  spending  100  dollars  more  than  -J 
of  his  income,  had  remaining  35  dollars  more  than  ^  of  it. 
Required  his  income.  u  \ 

Prob.  29.  In  the  composition  of  a  quantity  of  gunpowder, 
The  nitre  was  10  Ibs.  more  than  f  of  the  whole, 
The  sulphur  4£  Ibs.  less  than  £  of  the  whole, 
The  charcoal  2  Ibs.  less  than  ^  of  the  nitre. 
What  was  the  amount  of  gunpowder  ?   fe '. 

Prob.  30.  A  cask  which  held  146  gallons,  was  filled  with 
a  mixture  of  brandy,  wine,  and  water.  There  were  15  gal- 
lons of  wine  more  than  of  brandy,  and  as  much  water  as  the 
brandy  and  wine  together.  What  quantity  was  there  of  each  ? 

/,  'f4f,-  v    . 


Art.  162. a.]  SIMPLE  EQUATIONS.  89 

Prob.  31.  Four  persons  purchased  a  farm  in  company  for 
4755  dollars ;  of  which  B  paid  three  times  as  much  as  A  ; 
C  paid  as  much  as  A  and  B ;  and  D  paid  as  much  as  C  and 
B.  What  did  each  pay  ?  ]  ^  ^/f 

Prob.  32.  It  is  required  to  divide  the  number  99  into  five 
such  parts,  that  the  first  may  exceed  the  second  by  3,  be  less 
than  the  third  by  10,  greater  than  the  fourth  by  9,  and  less 
than  the  fifth  by  16.  jf_: 

Prob.  33.  A  father  divided  a  small  sum   among  four  sons. 

The  third  had  9  shillings  more  than  the  fourth  ; 

The  second  had  12  shillings  more  than  the  third  ; 

The  first  had  18  shillings  more  than  the  second  ; 

And  the  whole  sum  was  6  shillings  more  than  7  times 
the  sum  which  the  youngest  received. 

What  was  the  sum  divided  ?     j  <^  -4 

Prob.  34.  A  farmer  had  two  flocks  of  sheep,  each  con- 
taining the  same  number.  Having  sold  from  one  of  these 
39,  and  from  the  other  93,  he  finds  twice  as  many  remaining 
in  the  one  as  in  the  other.  How  many  did  each  flock  ori- 
ginally contain  ? 

Prob.  35.  An  express,  travelling  at  the  rate  of  60  miles 
a  day,  had  been  dispatched  5  days,  when  a  second  was  sent 
after  him,  travelling  75  miles  a  day.  In  what  time  will  the 
one  overtake  the  other  ? 

Prob.  36.  The  age  of  A  is  double  that  of  B,  the  age  of  B 
triple  that  of  C,  and  the  sum  of  all  their  ages  140.  What  is 
the  age  of  each  ? 

Prob.  37.  Two  pieces  of  cloth,  of  the  same  price  by  the 
yard,  but  of  different  lengths,  were  bought,  the  one  for  ,£5, 
the  other  for  «£6£.  If  10  be  added  to  the  length  of  each, 
the  sums  will  be  as  5  to  6.  Required  the  length  of  each 

piece.  ;       ;     .^ 


90  ALGEBRA.  [Sect.  VII. 

)  jt  y  *    * 

Prob.  38.  ^4.  and  B  began  trade  with  equal  sums  of  money. 
The  first  year,  A  gained  forty  pounds,  and  B  lost  40.  The 
second  year,  A  lost  £  of  what  he  had  at  the  end  of  the  first, 
and  B  gained  40  pounds  less  than  twice  the  sum  which  A 
had  lost.  B  had  then  twice  as  much  money  as  A.  What 
sum  did  each  begin  with  ? 

Prob.  39.  What  number  is  that,  which  being  severally 
added  to  36  and  52,  will  make  the  former  sum  to  the  latter, 
as  3  to  4  ? 

Prob.  40.  A  gentleman  bought  a  chaise,  horse,  and  har- 
ness, for  360  dollars.  The  horse  cost  twice  as  much  as  the 
harness  ;  and  the  chaise  cost  twice  as  much  as  the  harness 
and  horse  together.  What  was  the  price  of  each  ? 

Prob.  41.  Out  of  a  cask  of  wine,  from  which  had  leaked 
^  part,  21  gallons  were  afterwards  drawn;  when  the  cask 
was  found  to  be  half  full.  How  much  did  it  hold  ? 

Prob.  42.  A  man  has  6  sons,  each  of  whom  is  4  years  older 
than  his  next  younger  brother ;  and  the  eldest  is  three  times 
as  old  as  the  youngest.  What  is  the  age  of  each  ? 

Prob.  43.  Divide  the  number  49  into  two  such  parts,  that 
the  greater  increased  by  6,  shall  be  to  the  less  diminished  by 
11,  as  9  to  2. 

Prob.  44.  What  two  numbers  are  as  2  to  3 ;  to  each  of 
which,  if  4  be  added,  the  sums  will  be  as  5  to  7  ? 

Prob.  45.  A  person  bought  two  casks  of  porter,  one  of 
which  held  just  3  times  as  much  as  the  other ;  from  each  of 
these  he  drew  4  gallons,  and  then  found  that  there  were  4 
times  as  many  gallons  remaining  in  the  larger,  as  in  the  other. 
How  many  gallons  were  there  in  each  ? 

Prob.  46.  Divide  the  number  68  into  two  such  parts,  that 
the  difference  between  the  greater  and  84,  shall  be  equal  to 
3  times  the  difference  between  the  less  and  40. 


Art.  163.]  INVOLUTION.  91 

Prob.  47.  Four  places  are  situated  in  the  order  of  the  let- 
ters A,  B,  C,  D.  The  distance"  from  A  to  D  is  34  miles. 
The  distance  from  A  to  B  is  to  the  distance  from  C  to  D  as 
2  to  3.  And  J  of  the  distance  from  A  to  B,  added  to  half 
the  distance  from  C  to  D,  is  three  times  the  distance  from  B 
to  C.  What  are  the  respective  distances  ? 

Prob.  48.  Divide  the  number  36  into  3  such  parts,  that  J 
of  the  first,  J  of  the  second,  and  J  of  the  third,  shall  be  equal 
to  each  other. 

Prob.  49.  A  merchant  supported  himself  3  years,  for  ,£50 
a  year,  and  at  the  end  of  each  year,  added  to  that  part  of 
his  stock  which  was  not  thus  expended,  a  sum  equal  to  one 
third  of  this  part.  At  the  end  of  the  third  year,  his  original 
stock  was  doubled.  What  was  that  stock  ? 

Prob.  50.  A  general  having  lost  a  battle,  found  that  he 
had  only  half  of  his  army+3600  men  left  fit  for  action  ;  J  of 
the  army+600  men  being  wounded  ;  and  the  rest,  who  were 
£  of  the  whole,  either  slain,  taken  prisoners,  or  missing.  Of 
how  many  men  did  his  army  consist  ? 


SECTION   VIII. 

INVOLUTION. 

ART.  163.  DEFINITIONS. — (1.)  When  a  quantity  is  multi* 
plied  into  itself,  the  product  is  called  a  power.  Thus  3X3 
=9  ;  and  dXd=dd.  The  9  and  dd  are  powers  of  3  and  d. 

(2.)  Powers  are  divided  into  different  orders  or  degrees, 
as  the  first,  second,  third,  fourth,  fifth  powers,  fyc.,  which 
are  also  called  the  square,  cube,  Mquadrate,  fyc. 

QUEST. — What  is  a  power?  How  are  powers  divided?  What  is 
the  second  power  called  ?  Third  ?  Fourth  ? 


92  ALGEBRA.  [Sect.  VIII. 

They  take  their  name  from  the  number  of  times  the  root,  or 
first  power,  is  used  as  a  factor  in  producing  the  given  power. 

The  original  quantity  is  called  the  first  power  or  root  of 
all  the  other  powers,  because  they  are  all  derived  from  it. 
Thus,  2X2=4,  the  square  or  second  power  of  2. 

2X2X2=8,  the  cube  or  third  power. 
2X2X2X2=16,  the  biquadrate  or  fourth  power,  &c. 
And  «X«=##?  the  second  power  of  a. 

aXaXa=aaa,  the  third  power. 
aXaXaXa=aaacii  the  fourth  power,  &c. 
(3.)  The  number  of  times  a  quantity  is  employed  as  a 
factor  to  produce  the  given  power,  is  generally  indicated  by 
a  figure  or  letter  placed  above   it  on  the  right  hand.     This 
figure  or  letter  is  called   the  index  or  exponent.     Thus  aX& 
is  written  a2  instead  of  aa ;  and  aXaXa—a3. 

The  index  of  the  first  power  is  1 ;  but  this  is  commonly 
omitted,  for  a*=a. 

Obser.  An  index  is  totally  different  from  a  co-efficient.  The  latter  shows 
how  many  times  a  quantity  is  taken  as  a  part  of  a  whole ;  the  former 
how  many  times  the  quantity  is  taken  as  a  factor.  Thus  4a=a-^-a-}-a 
-f-a;  but  a*=aX«X«Xa=rt##a.  If  a=4,  then  4a=16;  and  a4  =256. 

(4.)  Powers  are  also  divided  into  direct  and  reciprocal. 

Direct  Powers  are  those  which  have  positive  indices,  as  df2, 
c?5,  &c.  and  are  produced  by  multiplying  a  quantity  into  itself. 
Thus  dXd=d2  ;  dXdXd=d3  ;  and  dXdXdXd=d*. 

A  Reciprocal  Poiver  of  a  quantity  is  the  quotient  arising 
from  dividing  a  unit  by  the  direct  power  of  that  quantity, 

as  i-  £•  i' &c-  <Art-  33-} 

QUEST. — From  what  do  they  take  their  name?  What  is  the  first 
power?  How  are  powers  denoted?  What  is  this  number  called? 
What  does  it  show  ?  What  is  the  difference  between  an  index  and  a 
co-efficient  ?  What  is  the  index  of  the  first  power  ?  Is  it  usually 
written  ?  How  else  are  powers  divided  ?  What  are  direct  powers  ? 
Reciprocal  powers  ? 


Arts.  164-166.]  INVOLUTION.  93 

It  is  produced  by  dividing  a  direct  power  by  its  root,  till 
we  come  to  the  root  itself ;  and  then  continuing  the  division, 

d3  d2 

we  obtain  the  reciprocal  powers.     Thus  -j=^2  ;  and  ~r— ^ 

and  -—1  ;  and  — r-d=—  ;  and  _-f-rf— — -,  &c. 
d  d  d2  d-  d3 

For  convenience  of  calculation,  reciprocal  powers  are 
written  like  direct  powers  with  the  sign  —  before  the  index  ; 

thus  — znj-2,  &c.     The  direct  and  reciprocal  powers  of  d, 

are  rf*,  d3,  d*,'*1,  d°,  d~l,  d~2,  d~3,  d~±,  &c. 

164.  INVOLUTION  is  the  process  of  finding  any  power  of  a 
quantity  by  multiplying  it  into  itself.     Hence, 

165.  To  involve  a  quantity  to  any  required  power. 
Multiply  the  quantity  into  itself,  till  it  is  taken  as  a  factor, 

as  many  times  as  there  are  units  in  the  index  of  the  power  to 
which  the  quantity  is  to  be  raised.     (Art.  80.) 

N.  B.  All  powers  of  1  are  the  same,  viz.  1.  For  IXlX 
lXl,-&c.=l. 

166.  A  single  letter  is  involved,  by  giving  it  the  index  of 
the  proposed  power ;  or  by  repeating  it  as  many  times,  as 
there  are  units  in  that  index. 

N.  B.  If  the  letter  or  quantity  has  a  co-efficient,  it  must  be 
raised  to  the  required  power  by  actual  multiplication. 

1.  The  4th  power  of  a,  is  a4  or  aaaa.  (Arts.  163,  165.) 

2.  The  6th  power  of  y,  is  y6  or  yyyyyy. 

.3.  The  wth  power  of  x,  is  xn  or  xxx . . .  n  times  repeated. 
4.  Required  the  3d  power  of  3x. 

QUEST. — How  are  reciprocal  powers  written  ?  What  is  involution  ? 
The  rule  ?  What  are  all  powers  of  1  ?  How  is  a  single  letter  involv- 
ed ?  If  the  quantity  has  a  co-efficient,  what  must  be  done  with  it  ? 


94  ALGEBRA.  Sect.  VIII. 

5.  Required  the  4th  power  of  4y. 

6.  Required  the  7th  power  of  2a. 

167.  The  method  of  involving  a  quantity  which  consists  of 
several  factors,  depends  on  the  principle,  that  the  power  of 
the  product  of  several  factors  is  equal  to  the  product  of  their 
powers. 

7.  Thus  (ay)2=a2y2.     For  by  Art.  164,  (ay)2—ayXay. 
But  ayXay—ayay—aayy—a2y2. 

8.  What  is  the  3d  power  of  bmx  ? 

9.  What  is  the  wth  power  of  ady  ? 

In  finding  the  power  of  a  product,  therefore,  we  may  either 
involve  the  whole  at  once ;  or  we  may  involve  each  of  the 
factors  separately,  and  then  multiply  their  several  powers 
into  each  other. 

10.  What  is  the  4th  power  of  dhy  > 

11.  What  is  the  3d  power  of  46  ? 

12.  What  is  the  nth  power  of  Gad  ? 

13.  What  is  the  3d  power  of  3wzX%  ? 

168.  SIGNS. —  When  the  root  is  positive,  all  its  powers  are 
positive  also  ;  but  when  the  root  is  negative,  the  ODD  powers 
are  negative,  while  the  EVEN  powers  are  positive.  (Art.  82.) 

169.  Hence  any  odd  power  has  the  same  sign  as  its  root. 
But  an  even  power  is  positive,  whether  its  root  is  positive  or 
negative.     Thus  -\-aX+a=a2.     And  — aX — a=a2. 

170.  To  involve  a  quantity  which  is  already  a  power. 
Multiply  the  index  of  the  quantity  into  the  index  of  the 

power  to  which  it  is  to  be  raised. 

QUEST. — On  what  principle  does  the  method  of  involving  a  quanti- 
ty which  consists  of  several  factors,  depend  ?  How  then  may  we  find 
the  power  of  a  product  ?  Rule  for  signs  ?  Does  this  differ  from  the 
rule  for  signs  in  multiplication  ?  What  sign  has  every  odd  power  ? 
Even  powers?  How  involve  a  quantity  which  is  already  a  power? 


Arts.  167-172.]  INVOLUTION.  95 

14.  The  3d  power  of  a2,  is  a2'*— a6. 

T?ora2=aa  :  and  the  cube  of  aa  is  aaXaaXaa=aaaaaa=aG; 
which  is  the  6th  power  of  a,  but  the  3d  power  of  a2. 

15.  Find  the  4th  power  of  a362. 

16.  Find  the  3d  power  of  4a2x. 

17.  Find  the  4th  power  of  2a*X3x2d. 

18.  Find  the  5th  power  of  (a+b)2. 

19.  Find  the  2d  power  of  («+&)". 

20.  Find  the  nth  power  of  (x— y)m. 
*      21.  Find  the  nth  power  of  (z+y)2. 

22.  Find  the  2d  power  of  (a3X&3). 

23.  Find  the  3d  power  of  (a*b2h*). 

171.  A  FRACTION  is  raised  to  a  power,  "by  involving  both 
the  numerator  and  the  denominator. 

a2 

24.  The  square  of  -  is  — .     For,  by  the  rule  for  the  mul- 
tiplication of  fractions,  ?Xr— 77=7^-     (Art.  1300 

bo      bo     b* 

25.  Find  the  2d,  3d,  and  nth  powers  of  -. 

a 

2xr2 

26.  Find  the  cube  of  -=— . 

% 

x2r 

27.  Find  the  nth  power  of . 

m 


28.  Find  the  square  of 


(x+1)3 

172.  A  compound  quantity  consisting  of  terms  connected 
by  -f-  and  — ,  is  involved  by  an  actual  multiplication  of  its 
several  parts.  Thus, 


• 


QUEST. — How  is  a  fraction  involved  ?     How  is  a  compound  quan- 
ity  involved  ? 


96  ALGEBRA.  [Sect.  VIII. 


29.  (a+b)1=a+b,       the  first  power 


a2+ab 
+ab+b2 


?      ...     the  second  power. 
a  +b 

ab2 


(a-f&)3—  a3+3«2H-3a&2+&3,  .     .  the  third  power.  ^ 
a  +  * 


—  a±+4«3&-f  6a262-|-4a63+6S  fourth  power. 

30.  Find  the  square  of  a  —  6. 

31.  Find  the  cube  of  a+1. 

32.  Find  the  square  of  a-{-b-{-h. 

33.  Required  the  cube  of  a 

34.  Required  the  4th  power  of 

35.  Required  the  5th  power  of  x+1. 

36.  Required  the  6th  power  of  1  —  b. 

173.  The  squares  of  binomial  and  residual  quantities  occur 
so  frequently  in  algebraic  processes,  that  it  is  important  to 
make  them  familiar. 

If  we  multiply  a-\-h  into  itself,  and  also  a  —  H  into  itself, 

37.  We  have  a+h  38.  And  a—  h 

a-\-h  a  —  h 

a2+ah 
-\-a7i+h2 


V    •. 


Arts.  173- 175.  J  INVOLUTION.  97 

Here  it  will  be  seen,  that  in  each  case,  the  first  and  last 
terms  are  sqqares  of  a  and  k ;  and  that  the  middle  term  is 
twice  the  product  of  a  into  h.  Hence  the  squares  of  bino- 
mial and  residual  quantities,  without  multiplying  each  of  the 
terms  separately,  may  be  found,  by  the  following  proposition.* 

The  square  of  a  binomial,  the  terms  of  which  are  both  posi- 
tive, is  equal  to  the  square  of  the  frst  term,  -\-  twice  the  pro- 
duct of  the  two  terms,  +  the  square  of  the  last  term. 

And  the  square  of  a  residual  quantity,  is  equal  to  the 
square  of  the  first  term,  — twice  the  product  of  the  two  terms, 
-f-  the  square  of  the  last  term. 

39.  Find  the  square  of  2a-}-b. 

40.  Find  the  square  of  h+l. 

41.  Find  the  square  of  ab^\-cd. 

42.  Find  the  square  of  6y-f3. 

43.  Find  the  square  of  3d— h. 

44.  Find  the  square  of  a — 1. 

174.  For  many  purposes,  it  will  be  sufficient  to  express  the 
powers  of  compound  quantities  by  exponents,  without  an  actu- 
al multiplication. 

45.  Thus  the  square  of  a-\-b,  is  a+6|2,  or  (a-\-b)2. 

46.  Find  the  nth  power  of  6c+8+x. 

In  cases  of  this  kind,  the  vinculum  must  be  drawn  over  all 
the  terms  of  which  the  compound  quantity  consists. 

175.  But  if  the  root  consists  of  several  factors,  the  vincu- 
lum which  is  used  in  expressing  the  power,  may  either  extend 
over  the  whole  ;  or  may  be  applied  to  each  of  the  factors 
separately,  as  convenience  may  require. 

QUEST. — What  is  life  square  of  a  binomial  whose  signs  are  plus  ? 
Of  a  residual  ?  Is  it  always  necessary  to  perform  the  multiplication  ? 
How  far  must  the  vinculum  extend  when  the  root  contains  factors? 

*  Euclid,  2.  4.  9 


98  ALGEBRA.  [Sect.  VIII, 

47.  Thus  the  square  of  (a+6)X(e+d),  is  either 

(a+b)X(c+d)\  or  (a+b)2X(c+d)2. 

For,  the  first  of  these  expressions  is  the  square  of  the  pro- 
duct of  the  two  factors,  and  the  last  is  the  product  of  their 
squares.  But  one  of  these  is  equal  to  the  other.  (Art.  167.) 

The  cube  of  aX(b+d),  is  aX(b+d)\  or  a3X(H-c/)3. 

176.  When  a  quantity  whose  power  has  been  expressed  by 
a  vinculum  and  an  index,  is  afterwards  involved  by  an  actual 
multiplication  of  the  terms,  it  is  said  to  be  expanded. 

48.  Thus  (a+6)2,  when  expanded,  becomes  a2-\-2ab-{-b2. 

49.  Expand  (a+b+h)2. 

BINOMIAL     THEOREM.* 

177.  To  involve  a  binomial  to  a  high  power  by  actual  mul- 
tiplication, as  in  Art.  172,  is  a  long  and  tedious  process.     A 
much  easier  and  more  expeditious  way  to  obtain  the  required 
power,  is  by  what  is  called  the  BINOMIAL  THEOREM.     This 
ingenious  and  beautiful  method  was  invented  by  SIR  ISAAC 
NEWTON,  and  has  been  deemed  of  so  great  importance  to 
mathematical  investigation,  that  it  is  inscribed  on  his  monu- 
ment in  Westminster  Abbey. 

178.  To  illustrate   this  theorem,  let  the  pupil  involve  the 
binomial  «+&,  (Art.  172,)  and  the  residual  a—  6,  to  the  2d, 
3d  and  4th  powers.     Thus, 


(a—  6)3—  a3—  3a2b+3ab2—  b3. 


QUEST.  —  What  is  meant  by  expanding  a  quantity?  What  is  the 
best  mode  of  involving  a  binomial  to  a  high  power?  Who  is  the  author 
of  this  theorem  ?  In  what  light  is  it  regarded  ?  What  is  (a-}-b)2  • 
(a+b)3?  («-f&)4  ?  (a-b)2  ?  (a—b)s?  (a—  i)4  ? 

*  See  Preface. 


Arts.  176-179.]  INVOLUTION.  99 

179.  By  a  careful  inspection  of  the  several  parts  of  the 
preceding  work,  the  following  particulars  will  be  observed  to 
be  common  to  each  power. 

I.  By  counting  the  terms  it  will  be  found  that  the  number 
in  each  power,  is  greater  by  1  than  the  index  of  that  power ; 
e.  g.  in  the  3d  power  the  number  of  terms  is  4  ;  in  the  4th 
power,  it  is  5,  &c. 

II.  If  we  examine  the  signs  we  shall  perceive  when  both 
terms  of  the  binomial  are  positive,  that  all  the  signs  in  every 
power  are  -f-  >  hut  when  the  quantity  is  a  residual,  all  the 
odd  terms,  reckoning  from  the  left,  have  the  sign  -)-,  and  all 
the  even  terms  have  the  sign  — .     Thus  in  the  4th  power, 
the  signs  of  the  first,  third  and  fifth  terms  are  -j-,  while 
those  of  the  second  and  fourth  are  — . 

III.  Let  us  now  direct  our  attention  to  the  indices. 

1.  It  will  be  seen  that  the  index  of  the  first  term,  or  the 
leading  quantity*  in  each  power,  always  begins  with  the  in- 
dex of  the  proposed  power,  and  decreases  1  in  each  succes- 
sive term  towards  the  right,  till  we  come  to  the  last  term  from 
which  the  letter  itself  is  excluded.     Thus  in  (adc^)4  the  in- 
dices of  the  leading  quantity  a,  are  4,  3,  2,  1. 

2.  The  index  of  the  following  quantity   begins  with  1  in 
the  second  term,  and  increases  regularly  by  1  to  the   last 
term,  whose  index  like  that  of  the  first,  is  the  index  of  the 
required  power.     Thus  in  (arfci)4  the  indices  of  the  follow- 
ing quantity  b,  are  1,  2,  3,  4. 

3.  We  shall  also  perceive,  that  the  sum  of  the  indices,  is 
the  same  in  each  term  of  any  given  power  ;    and  this  sum  is 

QUEST. — How  many  terms  are  there  in  each  power  ?  What  signs 
has  a  binomial?  Residual?  What  are  the  indices  of  the  leading 
quantity  ?  Of  the  following  quantity  ?  Which  is  the  leading  quantity? 
To  what  is  the  sum  of  the  indices  in  each  term  equal  ? 

*  The  first  letter  of  a  binomial,  is  called  the  leading  quantity,  an4 
the  other  the  following  quantity, 

* 


100  ALGEBRA.  [Sect.  VIII. 

equal  to  the  index  of  that  power.     Thus  the  sum  of  the  indi- 
ces in  each  of  the  terms  of  the  4th  power,  is  4. 

IV.  The  last  thing  to  be  considered  is  the  co-efficients  of 
the  several  terms. 

1.  The  co-efficient  of  the  first  and  last  terms  in  each  power, 
is  1 ;  the  co-efficient  of  the  second  and  next  to  the  last  terms, 
is  the  index  of  the  required  power.     Thus  in  the  3d  power, 
the  index  of  the  second  and  next  to  the  last  terms,  is  3  ;  and 
in  the  same  terms  in  the  4th  power,  it  is  4,  &c. 

2.  It  will  be  observed  also,  that  the  co-efficients  increase 
in  a  regular  manner  through  the  first  half  of  the  terms  ;  and 
then  decrease  at  the  same  rate  through  the  last  half.     Thus, 

in  the  4th  power  they  are          1,      4,       6,       4,    1, 
in  the  6th  power  they  are    1,    6,    15,     20,     15,    6,    1. 

3.  The  co-efficients  of  any  two  terms  equally  distant  from 
the  extremes,  are  equal  to  each  other.     Thus  in   the   4th 
power,  the  second  term  from  each  extreme  is  4 ;  in  the  6th 
power,  the  second  term  from   each  extreme  is  6,  and  the 
third  is  15. 

4.  The  sum  of  all  the  co-efficients  in  each  power,  is  equal 
to  the  number  2  raised  to  that  power.    Thus  (2)4i=:16 ;  also, 
the  sum  of  the  co-efficients  in  the  4th  power,  is  16,  and  (2)6 
=64  ;  so  the  sum  of  the  co-efficients  in  the  6th  power,  is  64. 

180.  If  we  involve  any  other  binomial  or  residual  to  any 
required  power  whatever,  and  examine  the  result,  we  shall 
find  the  foregoing  principles  are  common,  and  will  apply  to 

QUEST.— What  is  the  co-efficient  of  the  first  and  last  term  ?  What 
of  the  second  and  next  to  the  last?  What  is  peculiar  to  the  first  half 
of  them  ?  To  the  last  half?  How  do  thole  equally  distant  from  the 
extremes  compare  ?  To  what  is  the  sum  qf  all  the  co-efficients  in  any 
power  equal  ?  What  is  said  as  to  the  eatjnt  of  the  foregoing  princi- 
ples ?  What  then  do  they  furnish  ? 


Arts.  180,  181.]  INVOLUTION.  101 

all  examples.  Hence  we  may  safely  conclude,  that  they  are 
universal  principles,  and  may  be  employed  in  raising  all 
binomials  to  any  required  power.  They  are  the  basis,  or 
elements,  of  what  is  called  the  Binomial  Theorem. 

181.  The  BINOMIAL  THEOREM  may  be  defined,  a  general 
method  of  involving  binomial  quantities  to  any  proposed 
power,  and  is  comprised  in  the  following 


GENERAL     RULE. 

I.  SIGNS. — If  both  terms  of  the  binomial  have  the  sign  +, 
all  the  signs  in  every  power  will  be  -f- ;    but  if  the  given 
quantity  is  a  residual,  all  the  odd  terms  in  each  power,  reck? 
oning  from  the  left,  will  have  the  sign  -]-,  and  the  even  terms 
the  sign  — . 

II.  INDICES. —  The   INDEX   of  the  Jirst    term   or   leading 
quantity,  must  always  be  the  index  of  the  required  power ; 
and  this  decreases  regularly  by  1   through  the  other  terms. 
The  index  of  the  following  quantity  begins  with   1   in  the 
second  term,  and  increases  regularly  by  1  through  the  others. 

III.  CO-EFFICIENTS. — The  co-efficient  of  the  Jirst  term  is 
1 ;  that  of  the  second  is  equal  to  the  index  of  the  power  ;  and 
universally,  if  the  co-efficient  of  any  term  be  multiplied  by  the 
index  of  the  leading  quantity  in  that  term,  and  divided  by 
the  index  of  the  following  quantity  increased  by  1,  it  will 
give  the  co-efficient  of  the  succeeding  term. 

IV.  The  number  of  terms  will  always  be  one  greater  than 
the  power  required. 

In  algebraic  characters,  the  theorem  is 

b+nX  *an'2b2,  &c. 


QUEST. — What  is  the  Binomial  Theorem  ?     What  is  the  rule  for  the 
Bigns?    For  the  indices?    For  the  co-efficients  ?    The  number  of  terms? 

9* 


102  ALGEBRA.  [Sect.  VIII. 

N.  B.  It  is  here  supposed,  that  the  terms  of  the  binomial 
have  no  other  co-efficients  or  exponents  than  1.  Other  bino- 
mials may  be  reduced  to  this  form  by  substitution.  (Art.  159.) 

1.  What  is  the  6th  power  of  x-{-y  ? 
The  terms  without  the  co-efficients,  are 

z6,  x5y,  x*y2,  z3^3,  z2y4,  zy5,  y6. 
And  the  co-efficients,  are 

6X5     15X4     20X3 
'     °'     ~2~'    ~~3~'    ™T~'     b' 
that  is  1,    6,      15,          20,          15,        6,     1. 

Prefixing  these  to  the  several  terms,  we  have  the  power 
required  ; 

x6+6x5y+l5x*y2+20xsy*+15x2y*+6xy*+y6.   Ans. 

2.  What  is  the  5th  power  of  (d+h)  ? 

3.  What  is  the  wth  power  of  (b-\-y)  ? 

Ans.  bn+Abn-*y+Bbn-2y2+Cbn-*y*+Dbn-*y*,  &c. 
That  is,  supplying  the  co-efficients  which  are  here  repre- 
sented by  -4,  .B,  C,  &c. 

X^V,  &c. 


4.  What  is  the  5th  power  of  z2-j-3y2  ? 

Substituting  a  for  z2,  and  b  for  3y2,   (Art.  159,)  we  have 


And  restoring  the  values  of  a  and 

(Z2^_3y2)5_a.10_|_15a.8y2_|_90x 

-f243y10. 

5.  What  is  the  6th  power  of  (3z 

6.  What  is  the  2d  power  of  (a  —  b)  ? 

7.  What  is  the  3d  power  of  (a  —  b)  ? 


Q,UEST.  —  Can  this  rule  be  applied  to  binomials  whose  co-efficients 
exceed  1  ?     How  ? 


Arts.  182,  183.]  INVOLUTION.  103 

8.  What  is  the  4th  power  of  (a — 6)  ? 

9.  What  is  the  6th  power  of  (x— y)  ? 

10.  What  is  the  nth  power  of  (a — 6)  ? 

182.  When  one  of  the  terms  of  a  binomial  is  a  unit,  it  is 
generally  omitted  in  the   power,  except  in  the  first  or  last 
term  ;  because  every  power  of  1  is  1,   (Art.  165,)  and  this 
when  it  is  a  factor,  has  no  effect  upon  the  quantity  with 
which  it  is  connected.     (Art.  70.) 

11.  Thus  the  cube  of  (z+1)  is  z3+3x2Xl+3xXl2+l*, 
Which  is  the  same  as  x3+3x2+3x+l. 

12.  What  is  the  4th  power  of  (a — 1)  ? 

The  insertion  of  the  powers  of  1  is  of  no  use,  unless  it  be 
to  preserve  the  exponents  of  both  the  leading  and  the  fol- 
lowing quantity  in  each  term,  for  the  purpose  of  finding  the 
co-efficients.  But  this  will  be  unnecessary,  if  we  bear  in 
mind,  that  the  sum  of  the  two  exponents,  in  each  term,  is 
equal  to  the  index  of  the  power.  (Art.  179,  3.)  So  that,  if 
we  have  the  exponent  of  the  leading  quantity,  we  may  know 
that  of  the  following  quantity,  and  v.  v. 

13.  What  is  the  6th  power  of  (1— y)  ? 

14.  What  is  the  nth  power  of  (1+z)  ? 

183.  The  binomial  theorem  may  also  be  applied  to  quan- 
tities consisting  of  more  than  two  terms.     By  substitution,  sev- 
eral terms  may  be  reduced  to  two,  and  when  the  compound 
expressions  are  restored,  such  of  them  as  have  exponents 
may  be  separately  expanded.     (Art.  159.) 

15.  What  is  the  cube  of  a+b+c  ? 

Substituting  h  for  (&+c)>  we  have  a-{-(b-\-c)=a-}-h. 

QUEST. — When  one  of  the  terms  of  a  binomial  is  a  unit,  how  pro- 
ceed ?  Can  the  binomial  theorem  be  applied  to  quantities  which  hare 
more  than  two  terms  ?  How  ? 


104  ALGEBRA.  [Sect.  VIII. 

And  by  the  theorem,  (a+h)3=  a3+3a2A+3a/i2+A3. 
That  is,  restoring  the  value  of  7t, 


The  last  two  terms  contain  powers  of  (b-}-c)  ;  but  these 
may  be  separately  involved. 

183.  a.  Binomials,  in  which  one  of  the  terms  is  a  fraction, 
may  be  involved  by  actual  multiplication  ;  or  by  reducing  the 
given  quantity  to  an  improper  fraction,  and  then  involving  the 
fraction  according  to  Art.  171. 

16.  Find  the  square  of  z+£  ;     and    x  —  £,  as  in  Art.  173. 


X2+$X  X2—%X 

+1*+*  — i*+i 


Or,  reduce  the  mixed    quantities    to   improper  fractions. 
Thus,         s-f"2—     3      i  an(*  s — *~~o — •    (^rt*  120.) 

/2x+l\2      4z2+4z+l               /2z— 1\       4*2— 4z+l 
( — d — )  z:::: d — —  '  ( — 9 — )  ~ aT — — * 

Q 

17.  Find  the  square  of  «+  o- 

O          4+   „, 

'•         •  •   'JU-    ' 

18.  Find  the  square  of  x — --. 

ii'ii'3:  * 

19.  Find  the   square  of f-3zy. 

w   ' 

6 

20.  Find   the  square  of \-2abc. 

-»*%•«  '     -^r^ 

QUEST. — How  involve  a  binomial  when  one  term  is  a  fraction  ? 


Arts.  183.«.-185.a.]        INVOLUTION.  105 

EXAMPLES     FOR     PRACTICE. 

1.  Expand  (x+y)3.       2.  Expand  (a-(-6)4. 

3.  Expand  (a—  6)6.       4.  Expand  (z+y)5. 

5.  Expand  (z—  y)8.       6.  Expand  (m-\-n)  7. 

7.  Expand  (a+b)9.       8.  Expand  (x+y)10. 

9.  Expand  (x—y)13.  10.  Expand  (a—  b)7. 

11.  Expand  (a+6)8.  12.  Expand  (2+z)5. 

13.  Expand  (a—  6x+c)3.  14.  Expand  («+36c)3. 

15.  Expand  (2a&—  z)4.  16.  Expand  (4ab—  5c)2. 

17.  Expand  (3z—  6#)3.  18.  Expand  (5«-(-3d)3. 

ADDITION     OF     POWERS. 

184.  It  is  obvious  that  powers  may  be  added,  like  other 
quantities,  by  writing  them  one  after  another  with  their  signs* 
(Art.  47.) 

1.  Thus  the  sum  of  a3  and  b2,  is  «3+62. 

2.  And  the  sum  of  a2—  bn  and  A5—  -d*,  is  a2—bn+h5—d*. 

185.  The  same  powers  of  the  same  letters  are  like  quanti- 
ti^  (Art.  28  ;)  hence  their  co-  efficients  ^may  be  added  or  sub- 
tracted, as  in  Arts.  50  and  51. 

3.  Thus  the  sum  of  2«2  and  3a2,  is  5«2. 

4.  5.  6.  7.  8. 

To 
Add 


185.«.  But  powers  of  different  letters,  and  different  powers 
of  the  same  letter,  are  unlike  quantities,  (Art.  28  ;)  hence  they 

QUEST.  —  What  is  the  general  method  of  adding  powers?  How  are 
the  same  powers  of  the  same  letters  added  ?  How  are  powers  of  differ- 
ent letters^  and  different  powers  of  the  same  letter,  added  ? 


106  ALGEBRA.  [Sect.  VIII. 

can  be  added  only  by  writing  them  down  with  their  signs. 
(Art.  55.) 

9.  The  sum  of  a2  and  a3,  is  a2-f-a3. 

It  is  evident  that  the  square  of  a,  and  the  cube  of  a,  are 
neither  twice  the  square  of  a,  nor  twice  the  cube  of  a. 

10.  The  sum  of  a*bn  and  3«5i6,  is  «3£n+3a5&6. 

186.  From  the  preceding  principles  we  deduce  the  fol- 
lowing 

GENERAL      RULE      FOR     ADDING     POWERS. 

I.  If  the  powers  are  like  quantities,  add  their  co-efficients, 
and  to  the  sum  annex  the  common  letter  or  letters  with  their 
given  indices. 

11.  If  the  powers  are  unlike  quantities,  they  must  be  added 
by  writing  them,  one  after  another,  without  altering  their 
signs. 

II.  Add  5x(a—  by+x(a—  b)3  to  2z(a—  6)3+10x(a—  &)3. 

12.  Add  3(z+y)4+5a2—  4(*+y)4  to 

13.  Add  a362-f-z6y4+a2&3  and  —x 

14.  Add  5«26c3,  3a26c3,  a2bc*  and 

15.  Add  3«3+6c2+5a3+26c2  and  a3+56c2  to  6a+ 

2bc2. 

16.  Add  %(xy  —  cm)G,  3(xy  —  cm)6,  —^(xy  —  cm)6   to 


SUBTRACTION     OF     POWERS. 

187.  RULE.  —  Subtraction  of  powers  is  performed  in  the 
same  manner  as  addition,  except  that  the  signs  of  the  subtra- 
hend must  be  changed  as  in  simple  subtraction.  (Art.  60.) 

Q,PEST.  —  General  rule  for  adding  powers  ?  How  are  powers  sub- 
tracted ? 


Arts.  186-188.a.]  INVOLUTION.  107 

1.  From  2a4  take  —  6a4.     Ans.  8a*. 

2.  From  —36"       3.  3A266         4.  asbn        5.  5(a—  A)6 
Take        46n  4A266  «36n  2(a—  A)6 

6.  From  6a(«+6)4  take  a(a+6)4. 

7.  From  17a2x3+5zy2  take  12a2x3  —  ixy2. 

8.  From  3«3(62—  8)3  take  a*(b2—  8)3. 

9.  From  a2b3+x3yi  take  abbQ—x2y3. 

10.  From  5(x3+#4)3—  3(a2—  63)5  take  —  3(a2—  &3)3+ 

4(x3+^)3. 

11.  From  2x(a—  i)3+3(a—  6)3  take  x(a—  6)3+3(a~6)3. 

12.  From  £(x+y)2+£(a+&)3  take  i(x+y)2+f(«+6)3. 


MULTIPLICATION     OF     POWERS. 

188.  Powers  may  be  multiplied,  like  other  quantities,  by 
writing  the  factors  one  after  another,  either  with,  or  without, 
the  sign  of  multiplication  between  them.  (Art.  72.) 

1.  The  product  of  a3  into  62  is  a362  ;  and  x'3  into  cT  is 


2.  Mult.  h2b~n  3.  3a6y2  4.  dh^x^  5.  a263y2 
Into  a4  —  2#  46y*  asb2y 

188.a.  If  the  quantities  to  be  multiplied  are  powers  of  the 
same  root,  instead  of  writing  the  factors  one  after  another, 
as  in  the  last  article,  we  may  add  their  exponents,  and  the 
sum  placed  at  the  right  hand  of  the  root,  will  be  the  product 
required. 

The  reason  of  this  may  be  illustrated  thus  : 

a2Xa3   is  cz2a3,  (Art.  188.)     But  a2=aa,  and  as=aaa. 

And  aaXaaa^aaaaa^a5,  (Art.  80.)  The  sum  of  the  ex- 
ponents 2+3,  is  also  5.  So  dmXdn—  dm+n. 

N.  B.  The  same  principles  hold  true  with  respect  to  all 
other  powers  of  the  same  root. 


108  ALGEBRA.  [Sect.  VIII. 

189.  Hence  we  deduce  the  following 

GENERAL     RULE     FOR     MULTIPLYING     POWERS. 

I.  Powers  of  the  same  root  may  be  multiplied  by  adding 
their  exponents.     (Art.  167.) 

II.  If  the  powers  have  co-efficients,  these  must  be  multiplied, 
together,  and  their  product  prefixed  to  the  common  letter  or 
letters. 

III.  Powers  of  different  roots  are  multiplied  by  writing 
them  one  after  another,  either  with,  or  without,  the  sign  of 
multiplication  between  them. 

Thus  a2XaG=a2+6=a8.    And  *3X*2X*=*3+2+1=s6. 

6.          7.  8.  9.  10. 

Mult.        4an        3z*         &2y3         a263y2          (b+k— yT 
Into          2art        2z3         b*y  azb2y  &+&— y 

11.  Mult.  xz+x2y-\-xy2+y*  into  z— y. 

12.  Mult.  4x2y+3xy—  1  into  2x2— x., 

13.  Mult.  x3+x— 5  into  2x2+x+l. 

190.  The  rule  is  equally  applicable  to  powers  whose  ex- 
ponents  are  negative,  i.  e.  to  reciprocal  powers. 

14.  Thus  a-2Xa-z=a-*.    That  is,  —  X = 


aa      aaa       aaaaa 

15.  Mult.  y~n  into  y~m  into  y"4. 

16.  Mult,  a"2  into  a'3  into  a'8. 

17.  Mult,  a"2  into  a3  into  — a"5. 

18.  Mult,  a""  into  am  into  — a2". 

19.  Mult.  y~2  into  y2  into  —y~ny~s. 

QUEST. — How  are  powers  of  the  same  root  multiplied?  Of  differ- 
ent roots  ?  When  the  powers  have  co-efficients,  what  must  be  done 
with  them  ?  Is  this  rule  applicable  to  reciprocal  powers  ? 


Arts.  189-192.]  POWERS.  109 

20.  If  «-|-6  be  multiplied  into  a — 6,  the  product  will  be 
a2— 62,  (Art.  86;)   that  is, 

191.  The  product  of  the  sum  and  difference  of  two  quan- 
tities, is  equal  to  the  difference  of  their  squares. 

This  is  an  instanca  of  the  facility  with  which  general  truths 

tare  demonstrated  in  algebra. 
If  the  sum  and  difference  of  the  squares  be  multiplied,  the 
product  will  be  equal  to  the  difference  of  the  fourth  pow- 
ers, &c. 

21.  Mult,  (a— y)  into  (a+y). 

22.  Mult.  (a2—y2)  into  (a2+y2). 

23.  Mult,  (a4 — y*)  into  («4+y4). 

24.  Mult.  a2+a4+a6  into  a2-—!. 

25.  Mult.  3«(x2— y3)3  into  2a(x2— y3)4. 

26.  Mutt.  J(a2+63)3  into  J(o2+63)2. 

27.  Mult,  a3— 62  into  as+62. 

28.  Mult.  x*+x2y+zy2+y*  into  x+y. 

29.  Mult,  a4— 2a3&+4rt262— 8a63+1664  into  a+26. 

30.  Mult.  a2+6  into  a2 — 8. 

DIVISION     OF     POWERS. 

192.  Powers  may  be  divided,  like  other  quantities,  by  re- 
jecting from  the  dividend  a  factor  equal   to  the  divisor ;   or 
by  placing  the  divisor  under  the  dividend,  in  the  form  of  a 
fraction.     Thus  the  quotient  of  a*b2  divided  by  62,  is  a3. 

1.  2.  3.  4. 

By         — 3a3         263          a2  (a—h-\-y)* 

QUEST. — What  is  the  product  of  the  sum  and  difference  of  two  quan- 
tities equal  to  ? 

10 


110  ALGEBRA.  [Sect.  VIII. 

5.  The  quotient  of  a5  divided  by  a3,  is  ^.      But  this   is 

equal  to  a2.     For,  in  the  series 

a+S  a+3,  a+2,  o+1,  a°,  a'1,  a"2,  «'3,  a'S  &c. 
if  any  term  be  divided  by  another,  the  index  of  the  quotient 
will  be  equal  to  the  difference  between  the  index  of  the  divi- 
dend  and  that  of  the  divisor. 


Thus  8«4«'=  -«2-     And 

aaa  a 

193.  Hence  we  deduce  the  following 

GENERAL     RULE     FOR     DIVIDING     POWERS. 

I.  A  power  may  be  divided  by  another  power  of  the  same 
root,  by  subtracting  the  index  of  the  divisor  from  that  of  the 
dividend. 

II.  If  the  divisor  and  dividend  have  co-efficients,  the  co- 
efficient of  the  dividend  must  be  divided   by  that  of  the  divi" 
sor.     (Art.  96.) 

III.  If  the  divisor  and  dividend  are  both  compound  quan- 
tities^ the  terms  must  be  arranged,  and  the  operation  conduc- 


ted in  the  same  manner,  as  in  simple  division  of  compound 
quantities.     (Art.  107.) 

6.  Thus  y^-^-yz—y^-z—y^.     That  is,  ^  —y. 

J \J 

7.  Divide  an+l  by  a.  8.  Divide  zn  by  xn. 

9.  10.  11.  12.  13 

Divide      y2m  b6  8an+m          «n+3 

By  ym  b3  4am  a2 

194.  The  rule  is  equally  applicable  to  reciprocal  powers. 



Q.UEST. — How  is  a  power  divided  by  another  power  of  the  same 
root?  If  they  have  co-efficients,  how  proceed?  When  the  divisor 
and  dividend  are  both  compound  quantities,  how  ?  Is  this  rule  appli- 
cable to  reciprocal  powers  ? 

°' 


Arts.  193-194.a.  ]  POWERS.  Ill 

Thus  the  quotient  of  «"5  by  a"3,  is  a"2. 

1  1  1         aaa       aaa        1 

That  is ; = X— r~ — = — • 

aaaaa      aaa      aaaaa       1        aaaaa     aa 

15.  Divide  — x"5  by  x~3. 

16.  Divide  A2  by  A'1. 

17.  Divide  6«n  by  2a'3. 

18.  Divide  ba3  by  a. 

19.  Divide  b3  by  65. 

20.  Divide  a4  by  a7. 

21.  Divide  (a3-}-y3)m  by  (a3-\-y3)". 

22.  Divide  (&+*)"  by  (b+x). 

Examples  of  compound  divisors  with  indices.     (Art.  105.) 

23.  Divide  a3+*3  by  a+x. 

24.  Divide  a4+4.r4  by  a2— 2ax+2x2. 

25.  Divide  x6 — 1  by  x — 1. 

26.  Divide  a4+2«36-|-2a262+a63  J)y  a3+a2b-\-ab2. 

27.  Divide  6s— 16c8  by  b2— 2c2. 

28.  Divide  a6 — a*x — a2x3+2x4  by  a4 — x3. 

29.  Divide  «4+4a36+6a2&24-4«63+64  by  a2+2a6+62. 

30.  Divide  8z3— #3  by  2x— y. 

31.  Divide  x3— 3ax2+3a2x— a3  by  x— a. 

32.  Divide  2y3— 19^2+26y— 16  by  y—8. 

33.  Divide  XG— 1  by  x-f-1. 

34.  Divide  4x4— 9x2+6x— 3  by  2x2+3x— 1. 

35.  Divide  «4-j-4a2i-{-364  by  a-\-2b. 

36.  Divide  a4—  «2z2-f  2a3x— a4  by  x2—  ax+a2. 

194.a.  A  regular  series  of  quotients  is  obtained,  by  divi- 
ding the  difference  of  the  powers  of  two  quantities,  by  the 
difference  of  the  quantities  or  roots.  Thus, 


112  ALGEBRA.  [Sect.  VIII. 

37.  Divide  (y2—a2)  by  (y— a).     Ans.  y+a. 

38.  Divide  (y3 — a3)  by  (y — a). 

39.  Divide  (y4— a4)  by  (y— a). 

40.  Divide  (y5 — a5)  by  (y — a). 

GREATEST  COMMON  MEASURE. 

195.  (1.)  A  common  measure  of  two  or  more  quantities,  is 
a  quantity  which  will  divide,  or  measure  them  without  a  re- 
mainder.    (Art.  30.)     Thus  2d  is  a  common  measure  of 
I2d,  6d,  Sd,  &c. 

(2.)  The  greatest  common  measure  of  two  or  more  quanti- 
ties, is  the  greatest  quantity  which  will  divide  these  quantities 
without  a  remainder.  Thus  6d  is  the  greatest  common  mea- 
sure of  I2d  and  ISdj  and  8  is  the  greatest  common  measure 
of  16,  24  and  32. 

195.cc.  To  find  the  greatest  common  measure  of  two  or 
more  quantities. 

Divide  one  of  the  quantities  by  the  other,  and  the  preceding 
divisor  by  the  last  remainder,  till  nothing  remains ;  the  last 
divisor  will  be  the  greatest  common  measure. 

196.  The  greatest  common  measure  of  two  quantities  is  not 
altered,  by  multiplying  or  dividing  either  of  them  by  any 
quantity  which  is  not  a  divisor  of  the  other,  and  which  con-    • 
tains  no  factor  which  is  a  divisor  of  the  other. 

The  common  measure  of  db  and  ac  is  a.  If  either  be 
multiplied  by  d,  the  common  measure  of  abd,  and  ac,  or  of 
ab  and  acd,  is  still  a.  On  the  other  hand,  if  db  and  acd  are 

Q,UEST. — What  is  a  common  measure  ?  What  the  greatest  common 
measure  ?  How  found  ?  How  is  it  affected  by  multiplying  or  dividing 
either  of  the  quantities  by  any  quantity  which  is  not  a  divisor  of  the 
other  ? 


Arts.  195-196.]          COMMON  MEASURE.  113 

the  given  quantities,  the  common  measure  is  a  ;  and  if  acd 
be  divided  by  d,  the  common  measure  of  ab  and  ac  is  a. 

Hence  in  finding  the  common  measure  by  division,  the 
divisor  may  often  be  rendered  more  simple,  by  dividing  it  by 
some  quantity  which  does  not  contain  a  divisor  of  the  divi- 
dend. Or  the  dividend  may  be  multiplied  by  a  factor,  which 
does  not  contain  a  measure  of  the  divisor. 

1.  Find  the  greatest  common  measure  of 
6a2-t-llax+3x2,  and  6a2+7ax—  3x2. 

6a2+7ax-3x2)6a2+ll«x+3x2(l 
7<zx—  3x2 


Dividing  by  2x)4ax+6x2 

2a+3x)6a2+7ax—  3x2(3a—  x 
6a2+9ax 

.  _     VJ__      w 

_2ax—  3x2 
—  2ax—  3x2 


After  the  first  division  here,  the  remainder  is  divided  by  2x, 
which  reduces  it  to  2a-{-3x.  The  division  of  the  preceding 
divisor  by  this,  leaves  no  remainder.  Therefore  2a-\-3x  is 
the  common  measure  required. 

2.  What  is  the  greatest  common  measure  of  a;3  —  62x,  and 


3.  What  of  cx+x2,  and  a2c+a2x  ? 

4.  What  of  3x3—  24x—  9,  and  2x3—  16z—  6? 

5.  What  of  «*—  64,  and  a5—  62a3  ? 

6.  What  of  x2  —  1,  and  xy-\-y  ? 

7.  What  of  x3—  a3,  and  x4—  a4  ? 

8.  What  of  a2—a&—  262,  and  a2—  3a6+262  ? 

10* 


114  ALGEBRA.  [Sect.  VIII. 

9.  What  of  a4— z4,  and  a3—  a2x—ax2+x*  ? 
10.  What  of  a3—  ab2,  and  a2_^2a6+62  ? 


FRACTIONS     CONTAINING     POWERS. 

197.  In  the  section  on  fractions,  the  following  examples 
were  omitted,  for  the  sake  of  avoiding  an  anticipation  of  the 
subject  of  powers. 

1.  Reduce  ^-^  to  lower  terms.     Ans.  —  -. 
5a4     Saaaa     baa        .  . 

For     =-=--  (Art-117-) 


2.  Reduce  -—  -. 

3z5 

3.  Reduce 

,    -n  j 

4.  Reduce 

5.  Reduce  -—  •  and  -  -  to  a  common  denominator. 

a3  a~4 

a2Xa~~4  is  a~2,  the  first  numerator.     (Art.  118.) 
«3X#~3  is  a°:=l,  the  second  numerator. 
«3X«~"4  is  a""1,  the  common  denominator. 

a-2  I 

The  fractions  reduced  are  therefore  —  -  and  —  -. 


. 

b.  iteauce  —  -  -  ana  —  -  to  a  common  denominator. 
5a3  a4 

3a?2  dx 

7.  MulUply   _   mto  —  . 

8.  Multiply   *±*   into  ?=*. 


Arts.  197-199.]  ROOTS.  115 

"54-1  J2 I 

9.  Multiply  —J--    into  — ; — . 
x2  x-4-a 

~b^  h — 3  fln 

10.  Multiply — -    into  and   — -. 

or2  x  y~* 

a4          a3 

11.  Divide  —  by  — . 


10     Dividp  C 

fi—  x±  t      x2—  or2 

IW      •    • 

-\z  rtivirfo  _ 

a2       by         a 

y        ,    f»Tr               ' 

.  A3— i       <p 

14.  Divide  — n—  by  — ^ 
a*  A 


SECTION  IX. 

ROOTS. 

ART.  198.  If  we  resolve  63,  or  666,  into  equal  factors,  viz. 
6,  6,  and  6,  each  of  these  equal  factors  is  said  to  be  a  root 
of  63.  So  if  we  resolve  27  into  any  number  of  equal  fac- 
tors, as  3X3X3,  each  of  these  equal  factors  is  said  to  be  a 
root  of  27.  And  when  any  quantity  is  resolved  into  any 
number  of  equal  factors,  each  of  those  factors  is  said  to  be 
a  root  of  that  quantity. 

199.  A  root  of  a  quantity,  then,  is  a  factor,  which  multi- 
plied into  itself  a  certain  number  of  times,  will  produce  that 
quantity. 

The  number  of  times  the  root  must  be  taken  as  a  factor,  to 
produce  the  given  quantity,  is  denoted  by  the  name  of  the  root. 

QUEST. — What  is  a  root  ? 


116  ALGEBRA.  [Sect.  IX. 

Thus  2  is  the  4th  root  of  16;  because  2X2X2X2=16, 
where  two  is  taken  four  times  as  a  factor,  to  produce  16. 
So  a3  is  the  square  root  of  a6  ;  for  a3  Xa3~«6.  (Art.  189.) 
Powers  and  roots  are  correlative  terms.     If  one  quantity 
is  a  power  of  another,  the  latter  is  a  root  of  the  former.     As 
63  is  the  cube  of  6,  so  b  is  the  cube  root  of  63. 

200.  There  are  two  methods  in  use,  for  expressing  the 
roots  of  quantities ;  one  by  means  of  the  radical  sign  \/,  and 
the   other  by  a  fractional  index.     The  latter  is  generally  to 
be  preferred  ;  but  the  former  has  its  uses  on  particular  occa- 
sions. 

When  a  root  is  expressed  by  the  radical  sign,  the  sign  is 
placed  before  the  given  quantity,  in  this  manner,  \/a. 
Thus  %/a  is  the  2d,  or  square  root  of  a. 
%/a  is  the  3d,  or  cube  root. 

201.  The  figure  placed  over  the  radical  sign,  denotes  the 
number  of  factor s^  into  which  the  given  quantity  is  resolved  ; 
i.  e.  the  number  of  times  the  root  must  be  taken  as  a  factor 
to  produce  the  given  quantity. 

Thus  ^/a2  shows  that  a2  is  to  be  resolved  into  two  factors, 
and  ^/a3,  into  three  factors;  and  y'a  into  n  factors. 

The  figure  for  the  square  root  is  commonly  omitted,  and 
the  radical  sign  is  simply  written  before  the  quantity,  thus 

202.  When  a  figure  or  letter  is  prefixed  to  the  radical  sign, 
without  any  character  between  them,  the  two  quantities  are 
to  be  considered  as  multiplied  together. 

QUEST. — How  many  ways  to  express  the  roots  of  quantities  ?  The 
first  ?  Second  ?  Which  is  preferred  ?  What  does  the  number  placed 
over  the  radical  sign  denote  ?  Is  the  number  used  in  denoting  the 
square  root  ?  When  a  figure  or  letter  is  prefixed  to  the  radical  sign, 
what  does  it  show  ? 


Arts.  200-205.]  BOOTS.  117 

Thus  2\/a,  is  2X\/a>  that  is,  2  multiplied  into  the  root  of 
a,  or  which  is  the  same  thing,  twice  the  root  of  a. 

And  z\/b,  is  xX\/6,  or  x  times  the  root  of  6. 

When  no  co-efficient  is  prefixed  to  the  radical  sign,  1  is 
always  understood  ;  \^a  being  the  same  as  l\/a,  that  is, 
once  the  root  of  a. 

203.  The  cube  root  of  a6  is  a2.     For  a2Xa2X<*2=a6. 
(Art.  199.) 

Here  the  index  is  divided  into  three  equal  parts,  and  the 
quantity  itself  resolved  into  three  equal  factors. 

The  square  root  of  a2  is  a1  or  a.     For  aX«=a2- 

By  extending  the  same  plan  of  notation,  fractional  indices 
are  obtained. 

Thus,  in  taking  the  square  root  of  a1  or  a,  the  index  1  is 
divided  into  two  equal  parts,  £  and  £ ;  and  the  root  is  a2. 

On  the  same  principle,  the  cube  root  of  a,  is  cF—^/a. 

1 
The  wth  root,  is  an=%/a,  &c. 

204.  Every  root,  as  well  as  every  power  of  1,  is  1.     (Art. 
165.)     For  a  root  is  a  factor,  which  multiplied  into  itself  will 
produce  the  given  quantity.     But  no  factor  except  1  can  pro- 
duce 1,  by  being  multiplied  into  itself. 

So  that  1",  1,  \/l,  -J/1,  &c.  are  all  equal. 

205.  Negative  indices  are  used  in  the  notation  of  roots,  as 
well  as  of  powers.     (See  Art.  163,  4.) 

Thus  4-=a~*;    J-  =  a""*j   -L  =  a  2. 

a?  aa  an 

QUEST. — When  none  is  prefixed,  what  is  understood  ?  What  is 
every  root  of  1  ?  Do  roots  ever  have  negative  indices  ? 


118  ALGEBRA.  [Sect.  IX. 

POWERS     OF     ROOTS. 

206.  In  the  preceding  examples  of  roots,  the  numerator 
of  the  fractional   index  has  been  a  unit.     There  is  another 
class  of  quantities,  the  numerators  of  whose  indices  are  greater 

2.       3. 

than  1,  as  63,  c4,  &c.  These  quantities  may  be  considered 
either  as  powers  of  roots,  or  roots  of  powers. 

N.  B.  In  all  instances,  when  the  root  of  a  quantity  is  de- 
noted by  a  fractional  index,  the  denominator,  like  the  figure 
over  the  radical  sign,  (Art.  201,)  expresses  the  root,  and  the 

numerator  the  power.  Thus  a?  denotes  the  cube  root  of  the 
first  power  of  a,  i.  e.  that  a  is  to  be  resolved  into  three  equal 

factors ;  for  a'6  Xtf^Xa5^^-  On  the  other  hand,  c¥  denotes 
the  third  power  of  the  fourth  root  of  c,  or  the  fourth  root  of 
the  third  power.  One  expression  is  equivalent  to  the  other. 

1.  What  is  c?  equal  to  ?  2.  What  is  z*  equal  to  ? 

3.  What  is  y^  equal  to  ?  4.  What  is  b*  equal  to  ? 

5.  Write  the  fifth  root  of  the  fourth  power  of  a. 

6.  Write  the  seventh  power  of  the  ninth  root  of  d. 

207.  The  value  of  a  quantity  is  not  altered,  by  applying 
to  it  a  fractional  index  whose  numerator  and  denominator  are 
equal. 

2.         2.         n 

Thus  a=a2— «a— a».  For  the  denominator  shows  that 
a  is  resolved  into  a  certain  number  of  factors  ;  and  the  nu- 
merator shows  that  all  these  factors  are  included  in  a» . 

QUEST. — What  is  meant  by  powers  of  roots  ?  What  does  the  de- 
nominator of  a  fractional  index  express  ?  What  the  numerator  ? 

Explain  xT;  also  i^,  cTtf,  yTinr.  When  the  numerator  and  denomi- 
nator are  equal,  how  does  the  index  affect  the  quantity  ?  How  sim- 
plify such  an  expression  ? 


Arts.  206-208.]  ROOTS.  119 

On  the  other  hand,  when  the  numerator  of  a  fractional 
index  becomes  equal  to  the  denominator,  the  expression  may 
be  rendered  more  simple  by  rejecting  the  index. 

Instead  of  a»,  we  may  write  a. 

207.a.  The  index  of  a  power  or  root  may  be  exchanged, 
for  any  other  index  of  the  same  value. 

.2  4 

Instead  of  a3,  we  may  put  a*. 

For  in  the  latter  of  these  expressions,  a  is  supposed  to  be 
resolved  into  twice  as  many  factors  as  in  the  former  ;  and 
the  numerator  shows  that  twice  as  many  of  these  factors  are  to 
be  multiplied  together.  Hence  the  value  is  not  altered. 

208.  From  the  preceding  article,  it  will  be  easily  seen, 
that  a  fractional  index  may  be  expressed  in  decimals. 

7.  Thus  a^z=aTTr,  or  a0*5  ;  that  is,  the  square  root  is  equal 
to  the  fifth  power  of  the  tenth  root. 

8.  Express  a?  in  decimals.         9.  Express  a^  in  decimals. 
10.  Express  a2  in  decimals.       11.  Express  a?  in  decimals. 

12.  Express  a  ¥  in  decimals. 

In  many  cases,  however,  the  decimal  can  be  only  an  ap- 
proximation to  the  true  index. 

13.  Thus  ota0-3  nearly,   or  tt^=ao.33333  more  nearly, 
In  this  manner,  the  approximation  may  be  carried  to  any 

degree  of  exactness  which  is  required. 

14.  Express  a?  in  decimals.       15.  Express  cT*    in  dec. 
N.  B.  These  decimal  indices  form  a  very  important  class 

of  numbers,  called  logarithms. 

Q,UEST. — What  is  the  effect  when  one  index  is  exchanged  for  an- 
other index  of  the  same  value  ?  Can  a  fractional  index  he  expressed 
in  decimals  ?  Can  it  be  expressed  exactly  by  decimals  in  all  cases  ? 
What  class  of  numbers  are  thus  found  ? 


120  ALGEBRA.  [Sect.  IX. 

EVOLUTION. 
f~"~ 

209.  The  process  of  resolving  quantities  into  equal  fac- 
tors, is  called  Evolution. 

In  subtraction,  a  quantity  is  resolved  into  two  parts. 

In  division,  a  quantity  is  resolved  into  two  factors. 

In  evolution,  a  quantity  is  resolved  into  equal  factors. 

Evolution  is  the  opposite  of  involution.  One  is  finding  a 
power  of  a  quantity,  by  multiplying  it  into  itself.  The  other 
is  finding  a  root,  by  resolving  a  quantity  into  equal  factors. 
A  quantity  is  resolved  into  any  number  of  equal  factors,  by 
dividing  its  index  into  as  many  equal  parts. 

210.  From  the  foregoing  principles  we  deduce  the  following 

GENERAL  RULE  FOR  EVOLUTION. 

I.  Divide  the  index  of  the  quantity  by  the  number  express- 
ing the  root  to  be  found. 

Or,  place  over  the  quantity  the  radical  sign  belonging  to 
the  required  root. 

II.  If  the  quantities  have  co-efficients,  the  root  of  these 
must  be  extracted  and  placed  before  the  radical  sign,   or 
quantity.     Thus, 

To  find  the  square  root  of  d* ,  divide  the  index  4  by  2,  i.  e. 

d*—d2.     So  the  cube  root  of  d6,  is  d^^d2. 


Obser.  From  the  manner  of  performing  evolution  it  is  evident,  that 
the  plan  of  denoting  roots  by  fractional  indices,  is  derived  from  the 
mode  of  expressing  powers  by  integral  indices.  (Art.  203.) 

Q.UEST. — What  is  evolution  ?  Into  what  are  quantities  resolved  in 
subtraction?  Into  what,  in  division  ?  Into  what,  in  evolution  ?  How 
is  a  quantity  resolved  into  any  number  of  equal  factors  ?  Rule  for 
evolution  ?  What  is  the  plan  of  denoting  roots  by  fractional  indices 
derived  from  ? 


Arts.  209-210.a.]  ROOTS.  121 

1.  Required  the  cube  root  of  a6.     Ans.  a3. 

2.  Required  the  cube  root  of  a  or  a1.     Ans.  cr  or  $/a. 

For  a^Xa^Xa^,  or  ^/aX^/aX^/a=a.    (Art.  199.) 

3.  Required  the  fifth  root  of  ab. 

4.  Required  the  rath  root  of  a2. 

5.  Required  the  seventh  root  of  2d — z. 

6.  Required  the  fifth  root  of  (a — z)3. 

7.  Required  the  cube  root  of  a2. 

8.  Required  the  fourth  root  of  a"1. 

9.  Required  the  cube  root  of  a5. 

10.  Required  the  nth  root  of  xm. 

11.  Required  the  third  root  of  a6. 

12.  Required  the  fourth  root  of  x8. 

13.  Required  the  second  root  of  zn. 

14.  Required  the  fifth  root  of  d3. 

15.  Required  the  8th  root  of  a3. 

2 10. a.  The  rule  in  the  preceding  article  may  be  applied  to 
every  case  in  evolution.  But  when  the  quantity  whose  root 
is  to  be  found,  is  composed  of  several  factors,  there  will  fre- 
quently be  an  advantage  in  taking  the  root  of  each  of  the 
factors  separately. 

This  is  done  upon  the  principle  that  the  root  of  the  pro- 
duct of  several  factors,  is  equal  to  the  product  of  their  roots. 

Thus  Vafc— \/aX \fb.  For  each  member  of  the  equa- 
tion if  involved,  will  give  the  same  power. 

When,  therefore,  a  quantity  consists  of  several  factors,  we 
may  either  extract  the  root  of  the  whole  together ;  or  we  may 

QUEST. — What  is  the  root  of  the  product  of  several  factors  equal  to  ? 

11 


122  ALGEBRA.  [Sect.  IX. 

find  the  root  of  the  factors  separately,  and  then  multiply  them 
into  each  other. 

16.  The  cube  root  of  zy,  is  either  (xy)1*  or  z^y^. 

17.  Required  the  fifth  root  of  3y. 

18.  Required  the  sixth  root  of  abh. 

19.  Required  the  cube  root  of  86. 

20.  Required  the  nth  root  of  xny. 

211.  The  root  of  a  fraction  is  equal  to  the  root  of  the  nu- 
merator divided  by  the  root  of  the  denominator. 

4  i       J 

21.  Thus  the  square  root  of  J=^T.     For  —X^r— ?• 

6     6*  6*     &i 

22.  Required  the  nth  root  of  ^. 

23.  Required  the  square  root  of  — . 

212.  SIGNS. — 1.  An  odd  root  of  any  quantity  has  the  same 
sign  as  the  quantity  itself. 

2.  An  even  root  of  an  affirmative  quantity  is  ambiguous. 
N.  B.  An  even  root  of  a  negative  quantity  is  impossible. 

213.  But  an  even  root  of  an  affirmative  quantity  may  be 
either  positive  or  negative.     For,  the  quantity  may  be  pro- 
duced from  the  one,  as  well  as  from  the  other.  (Art.  169.) 

Thus  the  square  root  of  a2  is  +«,  or  — a. 
An  even  root  of  an  affirmative  quantity  is,  therefore,  said 
to  be  ambiguous,  and  is  marked  with  the  sign  -4-.     Thus  the 

square  root  of  36,  is  ±\/3b.     The  4th  root  of  x,  is  -4-x^. 
The  ambiguity  does  not  exist,  however,  when  from  the 
nature  of  the  case,  or  a  previous  multiplication,  it  is  known 

QUEST. — What  is  the  root  of  a  fraction   equal  to  ?     Rule  for  signs  ? 
What  is  the  even  root  of  a  negative  quantity  ? 


tarts.  211-216.]  ROOTS.  123 

(whether  the  power  has  actually  been  produced  from  a  positive 
jor  from  a  negative  quantity. 

214.  But  no  even  root  of  a  negative  quantity  can  be  found. 
The  square  root  of  — a2  is  neither  -\-a  nor  — a. 

For  +aX+a=+a*.     And  —  aX  —  a=+a2  also. 
An  even  root  of  a  negative  quantity  is,  therefore,  said  to  be 
impossible  or  imaginary. 

215.  The  methods  of  extracting  the  roots  of  compound 
quantities  are  to  be  considered  in  a  future  section.     But  there 
is  one  class  of  these,  the  squares  of  binomial  and  residual 
quantities,  which  it  will  be  proper  to  attend  to  in  this  place. 
The  square  of  a-\-b,  for  instance,  is  a2 -\-2ab-\-b2 ,  two  terms 
of  which,  a2  and  62,  are  complete  powers,  and  2ab  is  twice 
the  product  of  a  into  6,  that  is,  the  root  of  a2  into  the  root  of  b2. 

Whenever,  therefore,  we  meet  with  a  quantity  of  this  de- 
scription, we  may  know  that  its  square  root  is  a  binomial ; 
and  this  may  be  found,  by  taking  the  root  of  the  two  terms 
which  are  complete  powers,  and  connecting  them  by  the 
sign  -f-.  The  other  term  disappears  in  the  root.  Thus,  to 
find  the  square  root  of  x2 -\-2xy-\-y2,  take  the  root  of  x2,  and 
the  root  of  y2,  and  connect  thcrji  by  the  sign  -j-.  The  bino- 
mial root  will  then  be  z-\-y. 

In  a  residual  quantity,  the  double  product  has  the  sign  — 
prefixed,  instead  of  +.  The  square  of  a — 6,  for  instance,  is 
a2 — 2ab-\-b2.  (Art.  173.)  And  to  obtain  the  root  of  a 
quantity  of  this  description,  we  have  only  to  take  the  roots  of 
the  two  complete  powers,  and  connect  them  by  the  sign  — . 
Thus  the  square  root  of  x2 — %xy-\-y2,  is  x — y.  Hence, 

216.  To  extract  the  square  root  of  a  binomial  or  residual. 
Take  the  roots  of  the  two  terms  which  are  complete  powers, 

and  connect  them  by  the  sign  which  is  prefixed  to  the  other  term. 

QUEST. — How  extract  the  square  root  of  a  binomial,  or  residual  ? 


124  ALGEBRA.  [Sect.  IX. 

1.  To  find  the  root  of  ar»+2z+l. 

The  two  terms  which  are  complete  powers,  are  x2  and  1. 
The  roots  are  ff  and  1.     (Art.  204.)     Then  z-f-1.  Ans. 

2.  Find  the  square  root  of  x2—  2x+l.     (Art.  173.) 

3.  Find  the  square  root  of  a 

4.  Find  the  square  root  of  «2 


I2 

5.  Find  the  square  root  of  a2-\-ab  \       . 

6.  Find  the  square  root  of  a2 

c       c 

217.  A  root  whose  value  cannot  be  exactly  expressed  in 
numbers,  is  called  a  SURD,  or  irrational  quantity. 

Thus  \/2  is  a  surd,  because  the  square  root  of  2  cannot  be 
expressed  in  numbers,  with  perfect  exactness. 
In  decimals,  it  is  1.41421356  nearly. 

218.  Every  quantity  which  is  not  a  surd,  is  said  to   be  ra- 
tional. 

219.  By  RADICAL  QUANTITIES  is  meant,  all  quantities  which 
are  found  under  the  radical  sign,  or  tvhich  have  a  fractional 
index. 

REDUCTION     OF     RADICAL     QUANTITIES. 

220.  CASE  I.  To  reduce  a  rational  quantity  to  the  form  of 
a  radical  without  altering  its  value. 

Raise  the  quantity  to  a  power  of  the  same  name  as  the 
given  root)  and  then  apply  the  corresponding  radical  sign  or 
index. 

QUEST.  —  What  is  a  surd?  What  a  rational  quantity?  What  are 
radical  quantities  ?  How  reduce  a  rational  quantity  to  the  form  of  a 
radical  ? 


Arts.  217-221.]        RADICAL  QUANTITIES.  125 

1.  Reduce  a  to  the  form  of  the  wth  root. 
The  wth  power  of  a  is  an.     (Art.  166.) 

Over  this,  place  the  radical  sign,  and  it  becomes  ^/an. 

It  is  thus  reduced  to  the  form  of  a  radical  quantity,  with- 

n 

out  any  alteration  of  its  value.     For  :£/«"—««=«. 

2.  Reduce  4  to  the  form  of  the  cube  root. 

3.  Reduce  3a  to  the  form  of  the  4th  root. 

4.  Reduce  $ab  to  the  form  of  the  square  root. 

5.  Reduce  3Xa — x  to  the  form  of  the  cube  root. 

6.  Reduce  a2  to  the  form  of  the  cube  root. 

N.  B.  In  cases  of  this  kind,  where  a  power  is  to  be  reduced 
to  the  form  of  the  wth  root,  it  must  be  raised  to  the  rath  power, 
not  of  the  given  letter,  but  of  the  power  of  the  letter. 

Thus  in  the  6th  example,  a6  is  the  cube,  not  of  a,  but  of  a2. 

7.  Reduce  a364  to  the  form  of  the  square  root. 

8.  Reduce  am  to  the  form  of  the  wth  root. 

221.  CASE  II.  To  reduce  quantities  which  have  different 
indices,  to  others  of  the  same  value  having  a  common  index. 

1.  Reduce  the  indices  to  a  common  denominator. 

2.  Involve  each  quantity  to  the  power  expressed  by  the  nu- 
merator of  its  reduced  index. 

3.  Take  the  root  denoted  by  the  common  denominator. 

9.  Reduce  a4  and  b^  to  a  common  index. 

1st.  The  indices  £  and  £  reduced  to  a  common  denomina- 
tor, are  •&  and  &.  (Art.  118.) 

2d.  The  quantities  a  and  b  involved  to  the  powers  expressed 
by  the  two  numerators,  are  a3  and  b2. 

QUEST. — How  reduce  quantities  which  have  different  indices  to  a 
common  index  ? 

11* 


126  ALGEBRA.  [Sect.  IX. 

3d.   The   root  denoted    by  the   common    denominator  is 

the  TVth.     The  answer,  then,  is  (a3)1^"  and  (62)T^. 

The  two  quantities  are  thus  reduced  to  a  common  index, 
without  any  alteration  in  their  values. 

For  by  Art.  207.a,  «*=ia&,  which  by  Art.  206,  ^(a3)1^. 

1       J!L  JL 

And  universally,  an=amn=(am)mn. 

-i  ?• 

10.  Reduce  a?  and  bx*  to  a  common  index. 

Ans.  a*  and    6z*  ora3^  and   &%**. 


1  11 

11.  Reduce  a3  and  bn.  12.  Reduce  xn  and  ym. 

13.  Reduce  2^  and  3^.     14.  Reduce  (a+b)2  and  (x—yfi. 

15.  Reduce  cfi  and  b^.  16.  Reduce  x%  and  5^. 

222.  CASE  III.  To  reduce  a  quantity  to  a  given  index. 

Divide  the  index  of  the  quantity  by  the  given  index,  place 
the  quotient  over  the  quantity,  and  set  the  given  index  over 
the  whole. 

This  is  merely  resolving  the  original  index  into  two  factors. 
(Art.  209.) 

17.  Reduce  a^  to  the  index  £. 

By  Art  135,'  H-i=*xf  =f  =*. 
This  is  the  index  to  be  placed  over  a,  which  then  becomes 

a*;  and  the  given  index  set  over  this,  makes  it  (dv*,  the 
answer. 

18.  Reduce  a2  and  x*  to  the  common  index    . 


2-^-1=2x3=6,  the  first  index. 
£-H=£X3=:f,  the  second  index. 


\ 

) 


i  S-  4- 

Therefore  (a6)3  and  (ar)3  are  the  quantities  required. 

QUEST.  —  How  reduce  a  quantity  to  a  given  index  ? 


Arts.  222, 223.]        RADICAL  QUANTITIES.  127 

19.  Reduce  4^  and  3F  to  the  common  index  |. 

20.  Reduce  x2  and  y4  to  the  common  index  £. 

21.  Reduce  cr  and  63  to  the  common  index  £. 

22.  Reduce  c2  and  d*  to  the  common  index  f . 

-  -  i 

23.  Reduce  am  and  bm  to  the  common  index  -^. 

24.  Reduce  a2,  b^  and  c*  to  the  common  index  -j^-. 

223.  CASE  IV.  To  reduce  a  radical  quantity  to  its  most  sim- 
ple terms ;  i.  e.  to  remove  a  factor  from  under  the  radical  sign. 

Resolve  the  quantity  into  two  factors,  one  of  which  is  an 
exact  power  of  the  same  name  with  the  root ;  find  the  root  of 
this  power,  and  prefix  it  to  the  other  factor,  with  the  radical 
sign  betiveen  them. 

This  rule  is  founded  on  the  principle,  that  the  root  of  the 
product  of  two  factors  is  equal  to  the  product  of  their  roots. 
(Art.  210.0.) 

It  will  generally  be  best  to  resolve  the  radical  quantity  into 
such  factors,  that  one  ojf  them  shall  be  the  greatest  power 
which  will  divide  the  quantity  without  a  remainder. 

N.  B.  If  there  is  no  exact  power  which  will  divide  the  quan- 
tity, the  reduction  cannot  be  made. 

25.  Remove  a  factor  from  \/8. 

The  greatest  square  which  will  divide  8  is  4. 

We  may  then  resolve  8  into  the  factors  4  and  2.  For 
4X2=8. 

The  root  of  this  product  is  equal  to  the  product  of  the  roots 
of  its  factors  ;  that  is  A/8=V4XV%- 

But  \/4z=2.  Instead  of  \/4,  therefore,  we  may  substitute 
its  equal  2.  We  then  have  2X\/2  or  2\/2. 

QUEST. — How  reduce  a  radical  quantity  to  its  simplest  terms. 


128  ALGEBRA.                                     [Sect.  IX. 

26.  Reduce  ^/a^x.  Ans.  */a2X*/x—  aX\fx—  a*/z. 

27.  Reduce  \/18.  28.  Reduce 

29.  Reduce      /°  30.  Reduce 


31.  Reduce  (a3—  a2bft.  32.  Reduce  (54a66)*. 

33.  Reduce  \/98a2x.  34.  Reduce  v 


224.  CASE  V.  To  introduce  a  co-efficient  of  a  radical  quan- 
tity under  the  radical  sign.  (Art.  220.) 

Raise  the  co-efficient  to  a  power  of  the  same  name  as  the 
radical  part  ,  then  place  it  as  a  factor  under  the  radical  sign. 

35.  Thus,  ayb—  V~tfb. 


For  a—  y^ora"".  (Art.  207.)    And 

36.  Reduce  a(x  —  6)3  to  the  form  of  a  radical. 

Ans.  (o's  —  «8fc)*. 

1  n  I       7)2r      \2 

37.  Reduce  2ab(2ab2)*.  38.  Reduce  |f    24.^2)   ' 
39.  Reduce  2\/2.                       40.  Reduce  4b$/c. 

EXAMPLES     FOR     PRACTICE. 

1.  Reduce  5\/6  to  a  simple  radical. 

2.  Reduce  £\/5a  to  a  simple  radical. 

3.  Reduce  5^  and  67  to  the  common  index  £. 

4.  Reduce  a2  and  a2  to  the  common  index  £. 

5.  Reduce  \/98  to  its  simplest  form. 

6.  Reduce  \/243  to  its  simplest  form. 

QUEST.  —  How  introduce  a  co-efficient  under  a  radical  sign  ? 


Arts.  224-225.a.]       RADICAL  QUANTITIES.  129 

7.  Reduce  ^/54  to  its  simplest  form. 

8.  Reduce  7\/80  to  its  simplest  form. 

9.  Reduce  9y81  to  its  simplest  form. 

10.  Reduce  \/x2-\-ax2  to  its  simplest  form. 

11.  Reduce  \^198a2x  to  its  simplest  form. 

12.  Reduce  \^x3  —  a2x2  to  its  simplest  form. 

ADDITION     OF     RADICAL     QUANTITIES. 

225.  It  may  be  proper  to  remark,  that  the  rules  for  addi- 
tion, subtraction,  multiplication  and  division  of  radical  quan- 
tities are  essentially  the  same,  and  are  expressed  in  nearly 
the  same  language,  as  those  for  addition,  subtraction,  multi- 
plication and  division  of  powers.  So  also  the  rules  for  invo- 
lution and  evolution  of  radicals,  are  similar  to  those  for  invo- 
lution and  evolution  of  powers.  Hence,  if  the  learner  has 
made  himself  thoroughly  acquainted  with  the  principles  and 
operations  relating  to  powers,  he  has  substantially  acquired 
those  pertaining  to  radical  quantities,  and  will  find  no  diffi- 
culty in  understanding  and  applying  them. 

£25.  «.  Whe»  radical  quantities  have  th&^ame  radical  part) 
and  are  under  the  same  radical  sign  or  index,  they  are  like 
quantities.  (Art.  28.)  Hence  their  rational  parts  or  co-effi- 
cients may  be  added  in  the  same  manner  as  rational  quanti- 
ties, (Art.  56,)  and  the  sum  prefixed  to  the  radical  part. 
Thus,  2v"6+3V6 

1.  Add  yay  to 

2.  Add  —  2\/«  to  5\/a. 

3.  Add  4zA*  to 


QUEST.  —  What  is  said  respecting  the  rules  for  addition,  subtraction, 
multiplication  and  division  ;  also  of  involution  and  evolution  of  radi- 
cals? How  are  radical  quantities  added,  when  the  radical  parts  are 
alike  f 


130  ALGEBRA.  [Sect.  IX. 


4.  Add  76A    to 

5.  Add  y*/b—h  to 

226.  If  the  radical  parts  are  originally  different,  they  may 
sometimes  be  made  alike,  by  the  rules  for  reduction  of  radi- 
cal quantities. 

6.  Add  \/8  to  \S5Q.     Here  the  radical   parts  are  not  the 
same.     But  by  the  reduction  in  Art.  223,  V'S—  2\/2,  and 
V50=5\/2.     The  sum  then  is  7/y/2. 

7.  Add  V166  to 

8.  Add  \fa2x  to 

9.  Add  (36a2y)*  to 

10.  Add  V18a  to  3<</2a. 

227.  But  if  the  radical  parts,  after  reduction,  are  different, 
or  have  different  exponents,  the  quantities  are  unlike,  (Art. 
28  ;)  hence  they  can  be  added  only  by  writing  them  one  after 
the  other  with  their  signs.     (Art.  55.) 

11.  The  sum  of  3*/b  and  2*/a,  is  3V&+2\/«. 

It  is  manifest  that  three  times  the  root  of  b,  and  twice  the 
root  of  a,  are  neither  five  times  the  root  of  b,  nor  five  times 
the  root  of  a,  unless  b  and  a  are  equal. 

12.  The  sum  of  %/a  and  */a,  is  %/a+fya. 

The  square  root  of  a,  and  the  cube  root  of  a,  are  neither 
twice  the  square  root,  nor  twice  the  cube  root  of  a. 

228.  From  the  preceding  principle  we  deduce  the  following 

GENERAL  RULE  FOR  ADDITION  OF  RADICALS. 

I.  If  the  radicals  are  like  quantities,  add  their  co-efficients, 
and  to  the  sum  annex  the  common  radical  parts. 

QUEST.—  If  they  are  originally  different,  how  can  they  be  made 
alike  ?  When  they  are  unlike  quantities,  how  add  them  ?  General  rule  ? 


Arts.  226-229.]         RADICAL  QUANTITIES.  131 

II.  If  the  radicals  are  unlike  quantities,  they  must  "be  added 
by  writing  them,  one  after  another,  without  altering  their 
signs.  (Art.  186.) 

EXAMPLES     FOS     PRACTICE. 


1.  Add  \/27  to  /v/48. 

2.  Add  /v/72  to  */l28. 

3.  Add  A/180  to  \/405. 

4.  Add  33/40  to  3/135. 

5.  Add  43/54  to  5 3/128. 

6.  Add  9V243  to  lO/v/363. 

7.  Add  /v/816  to  \/496. 

8.  Add  \/9a2<f  to  \/16a2d. 

9.  Add  aV25x2c  to 

10.  Add  33/^6  to 


SUBTRACTION  OF  RADICAL  QUANTITIES. 

229.  RULE.  —  Subtraction  of  radical  quantities  is  to  "be  per- 
formed in  the  same  manner  as  addition,  except  that  the  signs 
in  the  subtrahend  are  to  be  changed  according  to  Art.  187. 
1.  From  \/ay  take  3\^ay.     Ans.  —  2\/«y. 


2.  From  4^/a+x  take  3$/a-\-x. 

3.  From  3A^  take  —  5A^ 

4.  From  a(x-\-y)*  take  b(x-}-y)*. 

5.  From  —  a~"  take  —  2a~  ". 

6.  From  \/50  take  >/8. 

7.  From  ?/64y  take 


Q.UEST.  —  How  are  radical  quantities  subtracted  ? 


132  ALGEBRA.  [Sect.  IX. 

8.  From  %/x  take  %/x. 

9.  From  2\/5Q  take  \/lS. 

10.  From  £/320  take  -£/40. 

11.  From  5\/20  take  3\/45. 


12.  From  \/8Qa*x  take 


MULTIPLICATION     OF     RADICAL     QUANTITIES. 

230.  Radical  quantities  may  be  multiplied,  like  other  quan- 
tities, by  writing  the  factors  one  after  another,  either  with  or 
without  the  sign  of  multiplication  between  them.     (Art.  72.) 

1.  Thus  the  product  of  \/«  into  */b,  ig  \/aX*/b. 

2.  The  product  of  $  into  y&,  is  J$y%. 

But  it  is  often  expedient  to  bring  the  factors  under  the  same 
radical  sign.'  This  may  be  done,  if  they  are  first  reduced  to 
a  common  index.  (Art.  221.) 

231.  Hence,  quantities  under  the  same  radical  sign  or  index, 
may  be  multiplied  together  like  rational  quantities,  the  pro- 
duct being  placed  under  the  common  radical  sign  or  index.* 
(Art.  210.) 

3.  Multiply  %/x  into  %/y,  that  is,  z*  into  y*. 

The  quantities  reduced  to  the  same  index,  (Art.  221,)  are 

(z3)*,  and  (y2)^,  and  their  product  is,  (xsy2)*=$/x*y2. 

4.  Multiply  \fa-\-m  into  \/a  —  m. 

5.  Multiply  \/dx  into  \/hy. 

_____  _  ____________________ 

QUEST.  —  How  may  radical  quantities  be  multiplied  ?  How  are  fac- 
tors brought  under  the  same  radical  sign?  How  multiplied  when  un- 
der the  same  radical  sign? 

*  The  case  of  an  imaginary  root  of  a  negative  quantity  may  be  con- 
sidered an  exception.  (Art.  214.) 


Arts.  230-232.]        RADICAL  QUANTITIES.  133 

6.  Multiply  cfi  into  x*. 

7.  Multiply  (a+yY  into  ~ 


8.  Multiply  am  into  zn. 

9.  Multiply  \fSxb  into  \^2xb.     Prod. 

In  this  manner  the  product  of  radical  quantities  often  be- 
comes rational. 

10.  Thus  the  product  of  \/2  into  \/18z=\/36=6. 

11.  Multiply  (a2y*)*  into  (a2y)%. 

232.  Roots  of  the  same  letter  or  quantity  may  le  multi* 
plied,  ly  adding  their  fractional  exponents. 

N.  B.  The  exponents,  like  all  other  fractions,  must  be  re- 
duced to  a  common  denominator,  before  they  can  be  united 
in  one  term.  (Art.  122.) 

12.  Thus  c$Xc$— a^=a^=a^. 

The  values  of  the  roots  are  not  altered,  by  reducing  their 
indices  to  a  common  denominator.  (Art.  207.a.) 


Therefore  the  first  factor  c$—a%  ) 
And  the  second  af^a5  * 


But  a=aXaX          (Art.  206.) 
And  a%=a*Xa*. 

The  product  therefore  is  a*Xa*Xa*Xa%X<fi=cft. 
N.  B.  In  all  instances  of  this  nature,  the  common  denom- 
inator of  the  indices  denotes  a  certain  root  ;  and  the  sum  of 

QUEST.  —  How  multiply  roots  of  the  same  letter?    How  are  expo- 
nents united  ? 

12 


134  ALGEBRA.  [Sect.  IX. 

the  numerators,  shows  how  often  this  is  to  be  repeated  as  a 

factor  to  produce  the  required  product. 

1 
JL       !        OL       —        ^^H: 

13.  Thus  anXam=amnXamn—amn. 

14.  Multiply  3y*  into  y*. 

15.  Multiply  (a+6)*  into  («+&)*. 

16.  Multiply  (a — y)^  into  (a — y)™. 

17.  Multiply  aT*  into  at*. 

18.  Multiply  y%  into  y~%. 

JL  — i 

19.  Multiply  «n  into  a  n. 

20.  Multiply  *"~*  into  x*~~. 

21.  Multiply  a2  into  a*. 

233.  Any  quantities  may  be  reduced  to  the  form  of  radi- 
cals, and  may  then  be  subjected  to  the  same  modes  of  ope- 
ration.    (Art.  220.) 

22.  Thus  yzXy^=y3+1^—y ~5~«       23.  And  xX%n—x  n  • 
The  product  will  become  rational,  whenever  the  numerator 

of  the  index  can  be  exactly  divided  by  the  denominator. 

24.  Thusa3X«*X«^«J¥2"=«4. 

25.  Multiply  (a+b)1*  into  («+&)""*. 

26.  Multiply  a5  into  d5. 

234.  When  radical  quantities  which  are  reduced  to  the 
same  index,  have  rational  co-efficients,  the  rational  parts  may 

QUEST. — Can  all  quantities  be  reduced  to  the  form  of  radicals  ? 
How  ?  How  may  they  be  treated  then  ?  When  radicals  have  co-effi- 
cients, what  must  be  done  with  them  ? 


Arts.  233-235.]        RADICAL  QUANTITIES.  135 

be  multiplied  together,  and  their  product  prefixed  to  the  pro- 
duct  of  the  radical  parts. 

27.  Multiply  a*/b  into  c\fd. 

The  product  of  the  rational  parts  is  ac. 
The  product  of  the  radical  parts  is  \/bd. 
And  the  whole  product  is  ac\fbd. 

28.  Multiply  ax^  into  b<fi. 

But  in  cases  of  this  nature  we  may  save  the  trouble  of  re- 
ducing to  a  common  index,  by  multiplying  as  in  Art.  230. 

29.  Thus  ax^  into  b$  is  mfibtfi. 

30.  Multiply  a(b+x$  into  y(b—  -xfi. 

31.  Multiply  a\fy2  into  b*/hy. 

32.  Multiply  a\/x  into 


33.  Multiply  a£~    into 

34.  Multiply  z^/3  into 

235.  If  the  rational  quantities,  instead  of  being  co-efficients 
to  the  radical  quantities,  are  connected  with  them  by  the 
signs  -f-  and  —  ,  each  term  in  the  multiplier  must  be  multi- 
plied into  each  in  the  multiplicand,  as  in  Art.  78. 

35.  Multiply  a+*/b 
Into          c+\/d 


Ans. 

36.  Multiply  «-(-vV  into  !+r\/y- 

a-{-<\/y-{-ar\/y-\-ry.    Ans. 

QUEST. — When  the  radicals  are  compound  quantities,  how  proceed  ? 


136  ALGEBRA.  [Sect.  IX. 

236.  Hence  we  deduce  the  following 

GENERAL  RULE  FOR  MULTIPLYING  RADICALS. 

I.  Radicals  consisting  of  the  same  letter  or  root,  are  mul- 
tiplied by  adding  their  fractional  exponents. 

II.  If  the  quantities  have  the  same  radical  sign,  or  index, 
multiply  them  together  as  you  multiply  rational  quantities, 
place  the  product  under  the  common  radical  sign,  and  to  this 
prefoc  the  product  of  their  co-efficients. 

III.  If  the  radicals  are  compound  quantities,  each  term  in 
the  multiplier  must  be  multiplied  into  each  term  of  the  multi- 
plicand by  writing  the  terms  one  after  another,  either  ivith,  or 
without,  the  sign  of  multiplication  between  them.    (Art.  189.) 

EXAMPLES     FOR     PRACTICE. 

1.  Multiply  \fa  into  ^/6. 

2.  Multiply  5\/5  into  3\/8. 

3.  Multiply  2\/3  into  3^/4. 

4.  Multiply  A/d  into  %/ab. 

5.  Multiply  \/-^r  into   \/  ~^- 

6.  Multiply  a(a—  xf  into  (c  —  d)X(axf. 

7.  Multiply  5x/8  into  3\/5. 

8.  Multiply  J\/6  into  T2T\/9. 

9.  Multiply  J\/18  into  5\/20. 
10.  Multiply  2\/3  into  1 


11.  Multiply  72Ja    into 

12.  Multiply  4+2v/2  into  2—  */2. 


QUEST.  —  General  rule  for  multiplying  radical  quantities  ? 


Arts.  236-238.]        BADICAL  QUANTITIES. 


DIVISION     OF     RADICAL     QUANTITIES. 

237.  The  division  of  radical  quantities  may  be  expressed, 
by  writing  the  divisor  under  the  dividend,  in  the  form  of  a 
fraction. 

1.  Thus  the  quotient  of  %/a  divided  by  \/&,  is  —-. 


2.  And  (a+h)*  divided  by  (b+xf  is 

(b+xf 

In  these  instances,  the  radical  sign  or  index  is  separately 
applied  to  the  numerator  and  the  denominator.  But  if  the 
divisor  and  dividend  are  reduced  to  the  same  index  or  radical 
sign,  this  may  be  applied  to  the  whole  quotient. 


3.  Thus  ya-±yb==  Z~.      For  the  root  of  a 

•v/6        v    b 

fraction  is  equal  to  the  root  of  the  numerator  divided  by  the 
root  of  the  denominator.     (Art.  211.) 

4.  Again,  ^/ab-^-^/b=i^/a.     For  the  product  of  this  quo- 
tient into  the  divisor  is  equal  to  the  dividend  ;  that  is, 

VaXV1>=Vd>'     Hence, 

238.  Quantities  under  the  same  radical  sign  or  index,  may 
be  divided  like  rational  quantities,  the  quotient  being  placed 
under  the  common  radical  sign  or  index. 

5.  Divide  (a;3^2)^  by  3^. 

These  reduced  to  the  same  index  are  (x3y2)^  and  (y2)  *. 
And  the  quotient  is  (z3)F—  z¥—  xa. 

QUEST.  —  How  is  the  division  of  radicals  expressed  ?  How  is  the 
radical  sign  to  be  placed  in  this  case  ?  How  divide  quantities  under 
the  same  radical  sign  ? 

12* 


138  ALGEBRA.  [Sect.  IX. 

6.  Divide  */6a3x  by  */3x. 

7.  Divide  Vdhx2  by  Vdx. 

8.  Divide  (a*+ax)^  by  a*. 

9.  Divide  (a3A)™  by  (axf. 

10.  Divide  (a2y2)*  by  (ay)%. 

239.  A  root  is  divided  by  another  root  of  the  same  letter 
or  quantity,  by  subtracting  the  index  of  the  divisor  from  that 
of  the  dividend. 

11.  Thus  a*-^o*= a*~*=a*~*=a*=fl*. 

For  a^=a*=a?Xd*Xa?,  and  this  divided  by  a*  is 


i* 

_l        1       1  _  1 

12.  In  the  same  manner,  am-~-an~i 

13.  Divide  (3a)^  by  'a*. 

14.  Divide  (ax)*  by  (ax)*. 

15.  Divide  a™  by  a™. 

2 

.    16.  Divide  (6-f^)n  by 

17.  Divide  (r2ys)r  by  (r2y3)T. 

239.a.  Powers  and  roote  of  the  same  letter,  may  also  be 
divided  by  each  other,  according  to  the  preceding  article. 

18.  Thus  a2-±a?=a2~:*=a?.     For  a*Xcfi= cF=a2. 

QUEST. — How  divide  one  root  by  another  root  of  the  same  letter  ? 
How  powers  and  roots  of  the  same  letter  ? 


Arts.  239-241.]         RADICAL  QUANTITIES.  139 

240.  When  radical  quantities  which  are  reduced  to  the 
same  index,  have  rational  co-efficients,  the  rational  parts  may 
be  divided  separately,  and  their  quotient  prefixed  to  the  quo- 
tient of  the  radical  parts. 

19.  Thus  ac\/bd-±-a\/b=:c\/d.     For  this  quotient  multi- 
plied into  the  divisor  is  equal  to  the  dividend. 

20.  Divide  24x\/ay  by  6/v/«- 

21.  Divide  18dh*/bx 


1  1 

22.  Divide  by(a*x2)n  by  y(ax)n. 

23.  Divide  16\/32  by  8\/4. 

24.  Divide  b\^xy  by  \fy. 

25.  Divide  a6(x26)*  by  a(zfi. 

These  reduced  to  the  same  index  are  ab(x2b)*  and  a(x2)*. 
The  quotient  then  is  b(b)%=(bbfi.     (Art.  224.) 
To  save  the  trouble  of  reducing  to  a  common  index,  the 

division  may  be  expressed  in  the  form  of  a  fraction. 

The  quotient  will  then  be  —^-  —  —  . 

«(*)* 
241.  Hence  we  deduce  the  following 

GENERAL     RULE     FOR     DIVIDING     RADICALS. 

I.  If  the  radicals  consist  of  the  same  letter  or  quantity,  sub- 
tract the  index  of  the  divisor  from  that  of  the  dividend,  and 
place  the  remainder  over  the  common  radical  part  or  root. 

II.  If  the  radicals  have  co-efficients,  the  co-efficient  of  the 
dividend  must  be  divided  by  that  of  the  divisor.     (Art.  96.) 

QUEST.  —  When  the  radicals  have  co-efficients,  what  is  to  be  done 
with  them  ?     General  rule  for  dividing  radical  quantities  ? 


140  ALGEBRA.  [Sect.  IX. 

III.  If  the  quantities  have  the  same  radical  sign  or  index, 
divide  them  as  rational  quantities,  and  place  the  quotient 
under  the  common  radical  sign.  (Art.  193.) 

EXAMPLES     FOR     PRACTICE. 


1.  Divide  2%/bc  by 

2.  Divide  10^/108  by  5^/4. 

3.  Divide  10^27  by  2*/3. 

4.  Divide  &v/108  by  2V6. 

5.  Divide  (a2b2d3)*  by  d*. 

6.  Divide  (  16a3—  I2a2x)^  by  2a. 

7.  Divide  6<v/  138  by  2\/6. 

8.  Divide  83/512  by  43/2. 

9.  Divide  J\/5  by  jV2< 

10.  Divide  \/7  by  3/7. 

11.  Divide  6/s/54  by  3^/2. 

12.  Divide  43/72  by  2  3/18. 

INVOLUTION     OF     RADICAL     QUANTITIES. 

242.  To  involve  a  radical  quantity  to  any  required  power. 
Multiply  the  index  of  the  root  into  the  index  of  the  power 
to  which  it  is  to  be  raised.     (Art.  170.) 


1.  Thus  the  square  of    a=«=«.  For  a 

2.  Required  the  cube  of  «T. 

1 

3.  Required  the  nth  power  of  am. 

4.  Required  the  fifth  power  of  <£  y%. 

QUEST.  —  How  are  radical  quantities  involved  ? 


Arts.  242-244.]         RADICAL  QUANTITIES.  141 

i  i 

5.  Required  the  cube  of  anxm. 

2     3 

6.  Required  the  square  of  crx^. 

6 

7.  Required  the  cube  of  «^. 

/T- 

8.  Required  the  wth  power  of  an. 

243.  A  root  is  raised  to  a  power  of  the  same  name,  T>y 
removing  the  index  or  radical  sign. 

When  the  radical  quantities  have  rational  co-efficients ,  these 
must  be  involved  by  actual  multiplication. 


9.  Thus  the  cube  of  Vb+x  is  6-f  x. 

_i 

10.  And  the  nth  power  of  (a  —  y)n  is  (a  —  y.) 

11.  The  square  of  a^/x  is  a21(/x2. 
For  a 


12.  Required  the  wth  power  of  amxm. 

13.  Required  the  square  of  a*/x  —  y. 

14.  Required  the  cube  of  3a  $/y. 

244.  But  if  the  radical  quantities  are  connected  with  others 
by  the  signs  -f-  and  —  ,  they  must  be  involved  by  a  multipli- 
cation of  the  several  terms,  as  in  Art.  172. 

15.  Required  the  squares  of  a+VV  anc* 


QUEST. — How  is  a  root  raised  to  a  power  of  the  same  name  ?  If 
the  radicals  have  co-efficients,  how  proceed  ?  If  the  radicals  are  com- 
pound quantities,  how  ? 


142  ALGEBRA.  [Sect.  IX. 

•   a 

16.  Required  the  cube  of  a  —  \fb. 

17.  Required  the  cube  of  2d-\-\^x. 

18.  Required  the  4th  power  of  \fd. 

19.  Required  the  4th  power  of  —  v  ax  —  1. 

_ 

20.  Required  the  6th  power  of  *</a-\-b. 

EVOLUTION     OF     RADICAL     QUANTITIES. 

245.  The  operation  for  finding  the  root  of  a  quantity  which 
is  already  a  root,  is  the  same  as  in  other  cases  of  evolution. 
Hence  we  derive  the  following 

RULE     FOR    THE     EVOLUTION     OF     RADICALS. 


I.  Divide  the  fractional  index  of  the  quantity  "by  the  num» 
ber  expressing  the  root  to  be  found.     Or, 

Place  the  radical  sign  belonging  to  the  required  root  over 
the  given  quantity. 

II.  If  the  quantities  have  rational  co-efficients,  the  root  of 
these  must  be  extracted,  and  placed  before  the  radical  sign, 
or  quantity.     (Art.  210.) 

1.  Thus,  the  square  root  of  a*,  is  «¥  *    zra^. 

2.  Required  the  cube  root  of  a(xy)*. 

3.  Required  the  nth  root  of  a\/by. 

4.  Required  the  4th  root  of  t/aX^/b. 

5.  Required  the  7th  root  of  128  \/d. 

245.a.  From  the  preceding  rules,  it  will  be  perceived  that 
powers  and  roots  may  be  brought  promiscuously  together, 
and  subjected  to  the  same  modes  of  operation. 

QUEST.  —  General  rule  for  the  evolution  of  radicals  ? 


Arts.  245,  245.a.]      RADICAL  QUANTITIES.  143 

^i,.-'.-'  -'T^iS-*'"* 

EXAMPLES     P'O  R     PRACTICE. 

1.  Find  the  4th  root  of  81#C~ 


-*^"-- 


2.  Find  the  6th  root  of  (a 

'    ' 

3.  Find  the  nth  root  of  (x— y)*. 

4.  Find  the  cube  root  of —  125«3x6.  - 

5.  Find  the  square  root  of  ,.  **    •. 

M  -r  *-  ti  *- 
\/Ju       U 

10 

6.  Find  the  5th  root  of  ^^"gfefc-' 

7.  Find  the  square  root  of  x2 — 66x+962.  ru  _ 

V2 

8.  Find  the  square  root  of  a2-\-ay  \       . 

9.  Reduce  ax2  to  the  form  of  the  6th  root.  (^ 

10.  Reduce  — Sy  to  the  form  of  the  cube  root. 

11.  Reduce  a2  and  a*  to  a  common  index. 

12.  Reduce  4^  and  54  to  a  common  index. 

'  13.  Reduce  cF  and  6*  to  the  common  index  J. 

14.  Reduce  2^  and  4^  to  the  common  index  J. 

15.  Remove  a  factor  from  \/294. 

16.  Remove  a  factor  from  V'x3 — a2x2. 

17.  Find  the  sum  and  difference  of  Vl6a2x  and  */4a2x. 

18.  Find  the  sum  and  difference  of  .J/192  and  .3/24. 

19.  Multiply  7£/18  into  5^4. 

20.  Multiply  4+2\/2  into  2— \/2.'  '^'\ 

21.  Multiply  a(a+\/c)^  into  &(«— \/c)^. 

t)-t       -  ^ 


144  ALGEBRA.  [Sect.  X. 

22.  Multiply  2(«+6)»  into  3(a+6)". 


23.  Divide  6\/54  by  3\/2. 

24.  Divide  4^72  by  2^/18. 


25.  Divide  */l  by 

26.  Divide  8^512  by  4v/2. 


27.  Find  the  cube  of 

28.  Find  the  square  of 

29.  Find  the  4th  power  of 

30.  Find  the  cube  of  A/X  —  */b. 


SECTION   X. 

DEDUCTION  OF  EQUATIONS  BY  INVOLUTION, 

ART.  246.  In  an  equation,  the  letter  which  expresses  the 
unknown  quantity  is  sometimes  found  under  a  radical  sign. 
We  may  have  \fx=La. 

To  clear  this  of  the  radical  sign,  let  each  member  of  the 
equation  be  squared,  that  is,  multiplied  into  itself.  We  shall 
then  have  \fxX*/x= aa.  Or,  (Art.  243,)  z— a2. 

The  equality  of  the  sides  is  not  affected  by  this  operation, 
because  each  is  only  multiplied  into  itself,  that  is,  equal  quan- 
tities are  multiplied  into  equal  quantities.  (Ax.  3.) 

The  same  principle  is  applicable  to  any  root  whatever. 
If  !j/x=a  ;  then  x=.an.  For  by  Art.  243,  a  root  is  raised  to 
a  power  of  the  same  name,  by  removing  the  index  or  radical 
sign.  Hence, 


Arts.  246,  247.]       QUADRATIC  EQUATIONS.  145 

247.  To  reduce  an  equation  when  the  unknown  quantity 
is  under  a  radical  sign. 

Involve  both  sides  to  a  power  of  the  same  name,  as  the  root 
expressed  by  the  radical  sign. 

N.  B.  It  will  generally  be  expedient  to  make  the  necessary 
transpositions,  and  to  clear  the  equation  of  fractions,  before 
involving  the  quantities  ;  so  that  all  those  which  are  not  under 
the  radical  sign  may  stand  on  one  side  of  the  equation. 


1.  Reduce  the  equation 
Transposing  -|-4 
Involving  both  sides, 

^-l-  4—9 
^/x—  9_  4—5 

2.  Reduce  the  equation 

a+^/x—  5=rf 

By  transposition 
By  involution 

^/z=rf-|-6  —  « 
x=(rf+6—  a)\ 

3.  Reduce  the  equation     £/x 

+1=4. 

4.  Reduce  the  equation     4-f- 

5.  Reduce  the  equation     \/a 
6.  Reduce  3+2\/x—  f  =6. 

7.  Reduce  4</  -=8.     *]£ 
v    o 

3/V/i^4=6+J. 

3-L.rf 

2_J_    /   —                ' 

+^x   V(«*+V*Y 

\f    *_»     ^     ,A      L-, 

/     ^-»  -4-J**^; 
/>-0 

;^<F 

8.  Reduce  (2x+3)*+4=7. 

:£*''ZZ2-    '     ^J't     ,  -i«^l 

9.  Reduce  */  \Z-\-x=%-\-\/y. 
10.  Reduce  ^x—a—^/x  —  J/> 
11.  Reduce  \/5X  Vx+2=2- 

r.^^^^'-^ 

#'r&'~~fi 

12.  Reduce  —  —  ^=^—  -i 
\/x          x 

^ 

QUEST.  —  Wlien   the   unknown  quantity  is  under  the  radical  sign, 
how  is  the   equation  reduced  ?     What  preparation  is  it  advisable  to 
make  before  involving  the  quantities  ? 
13 


15.  Reduce  x+V^+x^. 

16.  Reduce  x+a=+(62+). ; 

17.  Reduce  V^+Vz=;7^.     -  ^  ^  *J    ' 

18.  Reduce  Vx — 32=16 — \/^-  •  Af* 

19.  Reduce  </4z+17:=2\/z+l. 


REDUCTION  OF  EQUATIONS  BY  EVOLUTION. 

248.  In  many  equations,  the  letter  which  expresses  the 
unknown  quantity  is  involved  to  some  power.     Thus, 

In  the  equation  x2=\6, 

We  have  the  value  of  the  square  of  #,  but  not  of  x  itself. 
If  the  square  root  of  both  sides  be  extracted, 

We  shall  have  x—^. 

The  equality  of  the  members  is  not  affected  by  this  reduc- 
tion. For  if  two  quantities  or  sets  of  quantities  are  equal, 
their  roots  are  also  equal. 

If  (z+a)r'=6+A,  then  x+a—^b+h.     Hence, 

249.  To  reduce  an  equation  when  the  expression  contain- 
ing the  unknown  quantity  is  a  power. 

Extract  the  root  of  both  sides  which  corresponds  with  the 
power  expressed  by  the  index  of  the  unknown  quantity. 

QUEST. — When  the  unknown  quantity  is  a  power,  how  is  the  equa- 
tion reduced  ? 


Arts.  248-250.]      QUADRATIC  EQUATIONS.  147 

1.  Reduce  the  equation  6-f-#2 — 8=7 
By  transposition                           z2=7-|-8 — 6=9 
By  evolution  z=4-/\/9=4-3. 

The  signs  -|-  and  —  are  both  placed  before  \/9,  because  an 
even  root  of  an  affirmative  quantity  is  ambiguous.  (Art.  212.) 

2.  Reduce  the  equation  5z2 —  30=z2-|-34 
Transposing,  &c.  x2—  16 

By  evolution  x  =4-4. 

x2  x2 

3.  Reduce  the  equation  a-\ — 7-=^ — -r« 

4.  Reduce  the  equation  a-\-dxn=lO — xn. 

250.  From  the  preceding  articles  it  will  be  easy  to  see, 
that  to  reduce  an  equation  containing  a  root  of  a  power, 
(Art.  206,)  requires  both  involution  and  evolution. 

5.  Reduce  the  equation  ^/z2=4 

By  involution  z2=43=64 

By  evolution  z  =4-<\/64= 

6.  Reduce  the  equation  VZTO — a=k — d. 

7.  Reduce  the  equation  (z-j-tf)"  — — • 


<*-«)• 

i 

8.  Reduce  the  equation 

9.  Reduce  the  equation 

10.  Reduce  the  equation 

11.  Reduce  the  equation 

12.  Reduce  the  equation  (3-j-v'329-}-\/.r)2:i=144. 

QUEST. — Why  are  the  signs  -j-  a"d  —  placed  before  the  root?     How 
is  an  expression  containing  a  root  of  a  power  reduced  ? 


148  ALGEBRA.  [Sect.  X. 

PROBLEMS. 

Prob.  1.  A  gentleman  being  asked  his  age,  replied,  "If 
you  add  to  it  10  years,  and  extract  the  square  root  of  the 
sum,  and  from  this  root  subtract  2,  the  remainder  will  be  6." 
What  was  his  age  ? 

By  the  conditions  of  the  problem,          */x-}-lQ — 2=6 
By  transposition,  */x-\-W=6-}-2=8 

By  involution,  z-|-10=82=64 

And  z=64— 10=54. 

Proof.    (Art.  161.)     V54+10— 2=6. 

Prob.  2.  If  to  a  certain  number  22577  be  added,  and  the 
square  root  of  the  sum  be  extracted,  and  from  this  163  be 
subtracted,  the  remainder  will  be  237.  What  is  the  number  ? 

Let  z=the  number  sought,  6=163 

«=22577,  (Art.  159,)  c=237 

By  the  conditions  proposed  V#-|-a — b=c 

By  transposition,  Avx-\-a=c-\-b 

By  involution,  £-)-#= 

And  x—  (c+b)2 

Restoring  the  numbers,  (Art.  35,)  x=  (237+1 63) 2—  22577 

That  is,  x=  1 60000—22577=  137423. 

Proof.  >/137423+22577— 163=237. 

251.  When  an  equation  is  reduced  by  extracting  an  even 
root  of  a  quantity,  the  solution  does  not  always  determine 
whether  the  answer  is  positive  or  negative.  (Art.  212.)  But 
what  is  thus  left  ambiguous  by  the  algebraic  process,  is  fre- 
quently settled  by  the  statement  of  the  problem. 


Art.  251.]          QUADRATIC  EQUATIONS.  149 

Prob.  3.  A  merchant  gains  in  trade  a  sum,  to  which  320 
dollars  bears  the  same  proportion  as  five  times  this  sum  does 
to  2500.  What  is  the  amount  gained  ?  |  -\  *?  ^  %  % 

Prob.  4.  The  distance  to  a  certain  place  is  such,  that  if  96 
be  subtracted  from  the  square  of  the  number  of  miles,  the 
remainder  will  be  48.  What  is  the  distance  ?  /  ;X 

Prob.  5.  If  three  times  the  square  of  a  certain  number  be 
divided  by  4,  and  if  the  quotient  be  diminished  by  12,  the 
remainder  will  be  180.  What  is  the  number  ?  /  (f 

Prob.  6.  What  number  is  that,  the  fourth  part  of  whose 
square  being  subtracted  from  8,  leaves  a  remainder  equal  to  4? 

Prob.  7.  What  two  numbers  are  those,  whose  sum  is  to  the 
greater  as  10  to  7 ;  and  whose  sum  multiplied  into  the  less 
produces  270  ?  Q 

Prob.  8.  What  two  numbers  are  those,  whose  difference  is 
to  the  greater  as  2  to  9,  and  the  difference  of  whose  squares 
is  128?  $  ^ 

Prob.  9.  It  is  required  to  divide  the  number  18  into  two 
such  parts,  that  the  squares  of  those  parts  may  be  to  each 
other  as  25  to  16.  /  Q 


Prob.  10.  It  is  required  to  drvTde  the  number  14  into  two 
such  parts,  that  the  quotient  of  the  greater  divided  by  the 
less,  may  be  to  the  quotient  of  the  less  divided  by  the  greater, 
as  16  to  9.  &  %Q 

Prob.  11.  What  two  numbers  are  as  5  to  4,  the  sum  of 
whose  cubes  is  5103  ?  ;j  /£%  /£, 

Prob.  12.  Two  travellers,  A  and  B,  set  out  to  meet  each 
other,  A  leaving  the  town  C,  at  the  same  time  that  B  left  D, 
They  travelled  the  direct  road  between  C  and  D  ;  and  on 
meeting,  it  appeared  that  A  had  travelled  18  miles  more 
than  B,  and  that  A  could  have  gone  B's  distance  in  15  j  days, 

13. 


150  ALGEBRA.  [Sect.  X. 

but  B  would  have  been  28  days  in  going  A's  distance.  Re- 
quired the  distance  between  C  and  D. 

Prob.  13.  Find  two  numbers  which  are  to  each  other  as  8 
to  5,  and  whose  product  is  360.  <^  fy  £4*^^f  f  C  s 

Prob.  14.  A  gentleman  bought  two  pieces  of  silk,  which 
together  measured  36  yards.  Each  of  them  cost  as  many 
shillings  by  the  yard,  as  there  were  yards  in  the  piece,  and 
their  whole  prices  were  as  4  to  1.  What  were  the  lengths 
of  the  pieces  ?  A- 

Prob.  15.  Find  two  numbers  which  are  to  each  other  as  3 
to  2  ;  and  the  difference  of  whose  fourth  powers  is  to  the  sum 
of  their  cubes,  as  26  to  7. 

Prob.  16.  Several  gentlemen  made  an  excursion,  each 
taking  the  same  sum  of  money.  Each  had  as  many  servants 
attending  him  as  there  were  gentlemen ;  the  number  of  dol- 
lars which  each  had  was  double  the  number  of  all  the  ser- 
vants, and  the  whole  sum  of  money  taken  out  was  3456  dol- 
lars. How  many  gentlemen  were  there  ?  f  \^ 

Prob.  17.  A  detachment  of  soldiers  from  a  regiment  being 
ordered  to  march  on  a  particular  service,  each  company  fur- 
nished four  times  as  many  men  as  there  were  companies  in 
the  whole  regiment ;  but  these  being  found  insufficient,  each 
company  furnished  three  men  more  ;  when  their  number  was 
found  to  be  increased  in  the  ratio  of  17  to  16.  How  many 
companies  were  there  in  the  regiment  ? 

AFFECTED     QUADRATIC     EQUATIONS. 

252.  Equations  are  divided  into  classes,  which  are  distin- 
guished from  each  other  by  the  power  of  the  letter  that  ex- 
presses the  unknown  quantity.  Those  which  contain  only 

QUEST. — Into  what  are  equations  divided  ? 


Arts.  252, 253.]      QUADRATIC  EQUATIONS.  151 

the  first  power  of  the  unknown  quantity  are  called  simple 
equations,  or  equations  of  the  first  degree.  Those  in  which 
the  highest  power  of  the  unknown  quantity  is  a  square,  are 
called  quadratic,  or  equations  of  the  second  degree ;  those  in 
which  the  highest  power  is  a  cube,  are  called  cubic,  or  equa- 
tions of  the  third  degree,  &c. 

Thus  x=ia-{-b,  is  an  equation  ofthejirst  degree. 

x2— e,  and  z2-{-ax=d, 
are  quadratic  equations,  or  equations  of  the  second  degree, 

x*=7t,  and  x*+ax2+bx=d, 

are  cubic  equations,  or  equations  of  the  third  degree. 
253.  Equations  are  also  divided  into  pure  and  affected  equa- 
tions. A  pure  equation  contains  only  one  power  of  the  un- 
known quantity.  This  may  be  the  first,  second,  third,  or  any 
other  power.  An  affected  equation  contains  different  powers 
of  the  unknown  quantity.  Thus, 

C  x2=d — 6,  is  a  pure  quadratic  equation. 

C  x2-\-bx=d,  an  affected  quadratic  equation. 

C  z3— 6 — c,  a  pure  cubic  equation. 

C  x3+ax2+&x=rA,  an  affected  cubic  equation. 
In  a.  pure  equation,  all  the  terms  which  contain  the  unknown 
quantity  may  be  united  in  one,  (Art.  185,)  and  the  equation, 
however  complicated  in  other  respects,  may  be  reduced  by 
the  rules  which  have  already  been  given.  But  in  an  affected 
equation,  as  the  unknown  quantity  is  raised  to  different  pow~ 
ers,  the  terms  containing  these  powers  cannot  be  united. 
(Art.  185.a.) 

QUEST. — What  are  those  called  which  contain  only  the  first  power 
of  the  unknown  quantity  ?  When  the  unknown  quantity  is  a  square 
what?  When  a  cube?  How  else  are  equations  divided  ?  What  is  a 
pure  equation  ? 


152  ALGEBRA.  [Sect.  X. 

254.  An  affected  quadratic  equation  is  one  which  contains 
the  unknown  quantity  in  one  term,  and  the  square  of  that  quan- 
tity in  another  term. 

The  unknown  quantity  may  be  originally  in  several  terms 
of  the  equation.  But  all  these  may  be  reduced  to  two,  one 
containing  the  unknown  quantity,  and  the  other  its  square. 

255.  It  has  already  been  shown  that  a  pure  quadratic  is 
solved  by  extracting  the  root  of  both  sides  of  the  equation. 
An  affected  quadratic  may  be  solved  in  the  same  way,  if  the 
member  which  contains  the   unknown  quantity  is  an  exact 
square. 

Thus  the  equation  x2-\-2ax-{-a2=b-{-h1  may  be  reduced  by 
evolution.  For  the  first  member  is  the  square  of  a  binomial 

quantity.     (Art.  173.)     And  its  root  is  z-j-a.     There  fore  J 

_> 
z-(-a=V6+A,  and  by  transposing  a, 

x= Vb+h— a. 

256.  But  it  is  not  often  the  case,  that   a  member  of  an 
affected  quadratic  equation  is  an  exact  square,  till  an  addi- 
tional term  is  applied,  for  the  purpose  of  making  the  required 
reduction. 

In  the  equation  x2 -\-2ax-b,  the  side  containing  the  un- 
known quantity  is  not  a  complete  square.  The  two  terms 
of  which  it  is  composed  are  indeed  such  as  might  belong  to 
the  square  of  a  binomial  quantity.  (Art.  173.)  But  one  term 
is  wanting.  We  have  then  to  inquire,  in  what  way  this  may 
be  supplied.  From  having  two  terms  of  the  square  of  a  bino- 
mial given,  how  shall  we  find  the  third  ? 

Of  the  three  terms,  two  are  complete  powers,  and  the 
other  is  twice  the  product  of  the  roots  of  these  powers,  or 

QUEST. — What  is  an  affected  equation  ?  How  is  a  pure  quadratic 
equatum  solved  ?  How  an  affected  quadratic,  when  it  is  an  exact 
square  ? 


Arts.  254-258.]         QUADRATIC  EQUATIONS.  153 

which  is  the  same  thing,  the  product  of  one  of  the  roots  into 
twice  the  other. 

In  the  expression  x2+2az,  the  term  2ax  consists  of  the 
factors  2a  and  x.  The  latter  is  the  unknown  quantity.  The 
other  factor  20  may  be  considered  the  co-efficient  of  the  un- 
known quantity  ;  a  co-efficient  being  another  name  for  a  fac- 
tor. (Art.  24.)  As  x  is  the  root  of  the  first  term  x2  ;  the 
other  factor  20  is  twice  the  root  of  the  third  term,  which  is 
wanted  to  complete  the  square.  Therefore  half  20  is  the 
root  of  the  deficient  term,  and  a2  is  the  term  itself. 

The  square  completed  is  x2-|-20x+02,  where  it  will  be 
seen  that  the  last  term  a2  is  the  square  of  half  20,  and  2a  is 
the  co-efficient  of  x,  the  root  of  the  first  term. 

In  the  same  manner,  it  may  be  proved,  that  the  last  term 
of  the  square  of  any  binomial  quantity,  is  equal  to  the  square 
of  half  the  co-efficient  of  the  root  of  the  first  term. 

257.  From  this  principle  is  derived  the  following 

METHOD  FOR  COMPLETING  THE  SQUARE. 

Take  the  square  of  half  the  co-efficient  of  the  first  power  of 
the  unknown  quantity  ,  and  add  it  to  loth  sides  of  the  equation. 

258.  It  will  be  observed  that  there  is  nothing  peculiar  in 
the  solution  of  affected  quadratics,  except  the  completing  of 
the  square.     Quadratic  equations  are  formed  in  the   same 
manner  as  simple  equations  ;    and  after  the  square  is  com- 
pleted, they  are  reduced  in  the  same  manner  as  pure  equations. 

1.  Reduce  the  equation  x2-{-6ax=b 

Completing  the  square,       x2-|-6ax-|-9a2—  9«2-f-& 

Extracting  both  sides,  (Art.  255,) 

And 


Here  the  co-efficient  of  x,  in  the  given  equation,  is  60. 

_________  _____ 

QUEST.  —  What  is  the  first  method  for  completing  the  square  ?   What 
is  there  peculiar  in  the  solution  of  quadratics  ? 


154  ALGEBRA.  [Sect.  X. 

The  square  of  half  this  is  9a2,  which  being  added  to  both 
sides  completes  the  square.  The  equation  is  then  reduced 
by  extracting  the  root  of  each  member,  in  the  same  manner 
as  in  Art.  249,  excepting  that  the  square  here  being  that  of  a 
binomial,  its  root  is  found  by  the  rule  in  Art.  216. 

2.  Reduce  the  equation          x2 — 8bx—h. 

3.  Reduce  the  equation          x2 -\-ax—b-\-h. 

a2      a2 

Completing  the  square,      x2-\-ax-\— —  =        \  b-\-h. 

4.  Reduce  the  equation          x2 — x=h — d. 

5.  Reduce  the  equation          x2-\-3x—d-{-6. 

6.  Reduce  the  equation          x2 — abx—db — cd. 

7.  Reduce  the  equation          x2-\-  —  =h. 

8.  Reduce  the  equation          x2 — -—7h. 

b 

259.  In  these  and  similar  instances,  the  root  of  the  third 
term  of  the  completed  square   is  easily  found,   because  this 
root  is  the  same  half  co-efficient  from  which  the  term  has 
just  been  derived.     (Art.  257.)     Thus  in  the  last  example, 

half  the  co-efficient  of  x  is  —-,  and  this  is  the  root  of  the 

third  term  — — . 
46  2 

260.  When  the  first  power  of  the  unknown  quantity  is  in 
several  terms,  these  should  be  united  in  one,  if  they  can  be 
by  the  rules  for  reduction  in  addition.     But  if  there  are  lite- 
ral co-efficients,  these   may  be  considered   as  constituting, 

QUEST. — How  do  you  know  what  the  root  of  the  third  term  of  the 
completed  square  is  ?  When  the  first  power  is  in  several  terms,  what 
is  to  be  done  ?  If  there  are  literal  co-efficients  what  ? 


Arts.  259-262.]         QUADRATIC  EQUATIONS.  155 

together,  a  compound  co-efficient  or  factor,  into  which  the 
unknown  quantity  is  multiplied. 

Thus  ax+lx+dx—(a+b+d)Xx.  (Art.  97.)  The  square 
of  half  this  compound  co-efficient  is  to  be  added  to  both  sides 
of  the  equation. 

9.  Reduce  the  equation        x2-\-3x-{-2x-\-x=.d 
Uniting  terms  x2-{-6x=d 

Completing  the  square     x2-\-6x-{-9=:9-\-d 
And  x— 


10.  Reduce  the  equation        x2-\-ax-{-bx=h 
By  Art.  120,  *'+(«+&)  X*=A 

Therefore  x2+(a+b)X*+ 


11.  Reduce  the  equation        x2-\-ax  —  x=b. 
\s 

261.  Before  completing  the  square,   the  known  and  un- 

f  known  quantities  must  be  brought  on  opposite  sides  of  the 
equation  by  transposition  ;  the  square  of  the  unknown  quan- 
tity must  also  WJ  positive,  and  it  is  preferable  to  make  it  the 
first  or  leading  term. 

12.  Reduce  the  equation  a-}-5x  —  36=3a:  —  a;2 
Transp.  and  uniting  terms     x2-{-2x=3b  —  a 
Completing  the  square  x2  -j-2a:-f-l—  l-j-36  —  a 
And  a;— 


a;        36 

13.  Reduce  the  equation  -  =  — r-^ 

£      x— |—  & 

262.  If  the  highest  power  of  the  unknown  quantity  has  a  co-ef' 
ficient,  or  divisor,  before  completing  the  square  it  must  be  freed 
from  these  by  multiplication  or  division.  (Arts.  149,  154.) 

Q,UEST. — Before  completing  the  square  what  preparations  is  it  expe- 
dient to  make?  If  the  highest  power  has  a  co-efficient  or  divisor, 
what  should  be  done  ? 


156  ALGEBRA.  [Sect.  X. 

14.  Reduce  the  equation  a?2+24a  —  6h—l2x  —  5x2 
Transp.  and  uniting  terms  6x2  —  12x—  6h—  24a 
Dividing  by  6,  x2—2x=  h—Aa 
Completing  the  square,  x2  —  2x-\-I—l-\-h  —  4a 
Extracting  and  transp.  x=l^-*v  \-\-h  —  4«. 

bx2 

15.  Reduce  the  equation  h-\-2x=d  --  . 

a 

263.  If  the  square  of  the  unknown  quantity  is  in  several 
terms,  the  equation  must  be  divided  by  all  the  co-efficients  of 
this  square.  (Art.  155.) 

16.  Reduce  the  equation  bx2-{-dx2  —  4z=b  —  h 

Dividing  by  b+d,  *2 


17.  Reduce  the  equation  az2-\-x=h-\-3x  —  x2. 

Given  ax2-{-bx=d,  to  find  x.  t 

If  this  equation  is  multiplied  by  4a,  and  if  b2  is  added  to 
both  sides,  it  will  become, 


the  first  number  of  which  is  a  complete  square  of  the  bino- 
mial 2ax-\-b. 

264.  From  the  foregoing  principle  is  deduced 

A   SECOND    METHOD    OF    COMPLETING    THE    SQUARE. 

Multiply  the  equation  by  4  times  the  co-efficient  of  the 
highest  power  of  the  unknown  quantity,  and  add  to  both  sides 
the  square  of  the  co-efficient  of  the  lowest  power. 

The  advantage  of  this  method  is,  that  it  avoids  the  intro- 
duction of  fractions,  in  completing  the  square. 

QUEST.  —  If  the  square  of  the  unknown  quantity  is  in  several  terms, 
how  proceed  ?  What  is  the  second  method  of  completing  the  square  ? 
What  advantage  has  this  method  ? 


Arts.  263,  264.]      QUADRATIC  EQUATIONS.  157 


DEMONSTRATION. 

1.  The  object  of  multiplying  the   equation  by  the  co-effi- 
cient  of  the  highest  power,  is  to  render  the  first  term  a  per- 
fect square  without  removing  its  co-efficient,  and  at  the  same 
time  to  obtain  the  middle  term  of  the  square  of  a  binomial. 
But  we  must  multiply  all  the  terms  of  the  .equation  by  this 
quantity  to  preserve   the  equality  of  its   members.     (Ax.  3.) 
The  equation  above  when  mult,  by  a  becomes  a2x2+abx=ad. 

That  the  first  term  will,  in  all  cases,  be  rendered  a  complete 
square  when  multiplied  by  its  co-efficient,  is  evident  from  the 
fact,  that  it  will  then  consist  of  two  factors,  each  of  which  is 
a  square,  viz.  x2,  and  the  square  of  its  co-efficient.  But  the 
product  of  the  squares  of  two  or  more  factors,  is  equal  to 
the  square  of  their  product.  (Art.  167.) 

2.  It  will  be  seen  that  one  term  is  still  wanting  in  the  first 
member,  in  order  to  make  it  the  square  of  a  binomial,  viz.  the 
square  of  the  last  term.     (Art.  173.) 

This  deficiency  may  be  supplied  by  adding  to  both  sides 
the  square  of  half  the  co-efficient  of  the  lowest  power,  as  in 
the  first  method  of  completing  the  square.  But  in  taking  half 
of  this  co-efficient,  the  learner  will  often  be  encumbered  with 
fractions  which  it  is  desirable  to  avoid.  Thus  in  the  equation 

above,  half  of  the  co-efficient  of  the  lowest  power  is  -,  the 

I2 
square  of  which  is  — -.     Adding  this  to  both  sides,  the  equa- 

b2  I2 

tion  will  become,    a2x2 '-\-abx-\-  —  —ad-\-  — ,  the  first  mem- 
ber of  which  is  a  complete  square  of  the  binomial,  ax-\-  ^. 

QUEST. — Why  is  the  equation  multiplied  by  the  co-efficient  of  the 
highest  power  ?  How  does  it  appear  that  this  will  make  the  first  term 
an  exact  square  ?  Why  multiply  the  equation  by  4? 

14 


158  ALGEBRA.  [Sect.  X. 

Now  it  is  obvious,  that  multiplying  the  equation  by  4,  is 
the  same  as  removing  the  denominator  4  from  the  third  term. 
Hence  multiplying  the  equation  by  4  will  avoid  the  introduc- 
tion of  fractions,  and  also  leave  the  square  of  the  whole  of  the 
co-efficient  of  the  lowest  power  to  be  added  to  both  sides  ac- 
cording to  the  rule. 

The  first  term  evidently  continues  to  be  a  square  after  it  is 
multiplied  by  4,  for  it  is  still  the  product  of  the  powers  of 
certain  factors.  (Art.  167.) 

3.  It  will  be  perceived  at  once,  that  the  second  term  is 
composed  of  twice  the  root  of  the  first  term  into  the  co-effi- 
cient of  the  last  term,  which  constitutes  the  middle  term  of  a 
binomial  square.  (Art.  173.) 

Obser.  It  is  manifest  from  the  preceding  demonstration,  that  multiply- 
ing by  4  is  not  a  necessary  step  in  completing  the  square,  but  is  resorted  to 
as  an  expedient  to  prevent  the  occurrence  of  fractions.  When  therefore 
the  co-efficient  of  the  lowest  power  is  an  even  number,  so  that  half  of 
it  can  be  taken  without  a  remainder,  we  may  simplify  the  operation 
by  multiplying  by  the  co-efficient  of  the  highest  power  alone,  and  ad- 
ding to  both  sides  the  square  of  half  the  co-efficient  of  the  lowest  power 
of  the  unknown  quantity. 

Take  the  equation,  7z2-f-40z=7lf-. 

Multiplying  by  7,  it  becomes  49z2-f-280z=500 

Adding  the  square  of  half  the  co-efficient,    49z2-f-280z-|-4 00=900 
By  evolution  and  transposition,  x=40. 

265.  From  the  preceding  principles  we  may  also  deduce 

OTHER  METHODS  OF  COMPLETING  THE  SQUARE. 

Multiply  the  equation  by  16  times  the  co-efficient  of  the 
highest  power  of  the  unknown  quantity,  and  add  to  both  sides 
4  times  the  square  of  the  co-efficient  of  the  lowest  power. 

QUEST. — Why  add  the  square  of  the  co-efficient  of  the  lowest  power 
to  both  sides  ?  How  may  the  operation  be  simplified  when  the  co- 
efficient of  the  lowest  power  is  an  even  number? 


Arts.  265, 265.a.]       QUADRATIC  EQUATIONS.  159 

And  universally,  multiplying  the  equation  by  the  product 
of  any  square  number,  as  n2,  into  the  co-efficient  of  the  high' 
est  power,  and  adding  to  both  sides  the  square  of  half  the 
root  of  this  number  into  the  square  of  the  co-efficient  of  the 
lowest  power,  will  render  it  a  complete  square. 
Take  the  equation  x2 — 3z— 4 

Multiplying  by  16,  &c.  16z2 — 48z-f-36=100 

By  evolution  and  transposition,       x=4 
Or,  take  the  equation  ax2-}-cx—d. 

Mult,  by  n2,  &c.  n2a2x2+n2acx+^-=n2ad+^- ;  the 
first  member  of  which  is  the  square  of  the  binomial,  naz-f-— -. 

4> 

There  is  an  obvious  advantage,  however,  in  employing  4  in 
preference  to  any  other  square  number.  For  multiplying  the 
equation  by  4  times  the  co-efficient  of  the  highest  power,  will 
produce  the  middle  term  of  a  binomial  square,  the  third  term 
of  which  is  the  square  of  the  co-efficient  of  the  lowest  power. 

18.  Reduce  the  equation         ax2-\-dx=^h. 

19.  Reduce  the  equation         3x2-\-5x=42. 

20.  Reduce  the  equation         x2 — 15zi=z — 54. 

265.a.  In  the  square  of  a  binomial,  the  first  and  last  terms  are 
always  positive.  For  each  is  the  square  of  one  of  the  terms 
of  the  root,  and  all  even  powers  are  positive.  (Arts.  168, 173.) 

If  then  — x2  occurs  in  an  equation,  it  cannot  with  this 
sign  form  a  part  of  the  square  of  a  binomial.  But  if  all 
the  signs  in  the  equation  be  changed,  whilst  the  equality  of  the 
sides  will  be  preserved,  the  term  — x2  will  become  positive, 
and  the  square  may  then  be  completed.  (Art.  146.) 

QUEST. — What  other  ways  of  completing  the  square  are  mentioned? 
In  the  square  of  a  binomial,  what  sign  have  the  first  and  last  terms  ? 
If  the  square  of  the  unknown  quantity  has  the  sign  —  before  it,  what 
must  be  done  ? 


160  ALGEBRA.  [Sect.  X. 

21.  Reduce  the  equation  — x2-}-^x=d — h 
Changing  all  the  signs  x2 — 2xr=A — d. 

22.  Reduce  the  equation  4x — x2= — 12. 

266.  In  a  quadratic  equation,  the  first  term  x2  is  the  square 
of  a  single  letter.     But  a  binomial  quantity  may  consist  of 
terms,  one  or  both  of  which  are  already  powers. 

Thus  x3+«  is  a  binomial,  and  its  square  is  x6 -\-2ax*  +a2, 
where  the  index  of  x  in  the  first  term  is  twice  as  great  as  in 
the  second.  When  the  third  term  is  deficient,  the  square 
may  be  completed  in  the  same  manner  as  that  of  any  other 
binomial.  For  the  middle  term  is  twice  the  product  of  the 
roots  of  the  two  others. 

So  the  square  of  o^+a,  is  x2n+2axn+a2. 

1  2.          J. 

And  the  square  of  #n-)-a,  is  xn-\-2axn-}-a2.     Therefore, 

267.  Any  equation  which  contains  only  two  different  pow- 
ers or  roots  of  the  unknown  quantity,  the  index  of  one  of 
which  is  twice  that  of  the  other ,  may  be  solved  in  the  same 
manner  as  a  quadratic  equation,  by  completing  the  square. 

N.  B.  It  must  be  observed,  that  in  the  binomial  root,  the 
letter  expressing  the  unknown  quantity  may  still  have  a  frac- 
tional or  integral  index,  so  that  a  farther  operation  may  be 
necessary.  (Art.  250.) 

23.  Reduce  the  equation  x* — x2~6 — a 
Completing  the  square  x- 
Extracting  and  transposing,  x2=j 
Extracting  again,  (Art.  249,)  x: 

24.  Reduce  the  equation  x2n — 46xn=«. 

QUEST. — How  solve  equations  which  contain  only  two  different 
powers  or  roots  of  the  unknown  quantity,  when  the  index  of  one  is 
twice  that  of  the  other  ? 


Arts.  266-269.]      QUADRATIC  EQUATIONS.  161 

25.  Reduce  the  equation  z-[-4\/z=A  —  «. 

2         JL 

26.  Reduce  the  equation  xn  -\-8zn=a-{-b. 

268.  The  solution  of  a  quadratic  equation,  whether  pure 
or  affected,  gives  two  results.     For  after  the  equation  is  re- 
duced,  it  contains  an  ambiguous  root.     In  a  pure  quadratic, 
this  root  is  the  whole  value  of  the  unknown  quantity. 

Thus  the  equation  z2=:64, 

Becomes  when  reduced,  a;—  ;t\/64.  (Art.  249.) 
That  is,  the  value  of  x  is  either  +8  or  —  8,  for  each  of 
these  is  a  root  of  64.  Here  both  the  values  of  x  are  the 
same,  except  that  they  have  contrary  signs.  This  will  be 
the  case  in  every  pure  quadratic  equation,  because  the  whole 
of  the  second  member  is  under  the  radical  sign.  The  two 
values  of  the  unknown  quantity  will  be  alike,  except  that 
one  will  be  positive,  and  the  other  negative. 

269.  But  in  affected  quadratics,  a  part  only  of  one  side  of 
the  reduced  equation  is  under  the  radical  sign.     When  this 
part  is  added  to,  or  subtracted  from,  that  which  is  without  the 
radical  sign  ;  the  two  results  will  differ  in  quantity,  and  will 
have  their  signs  in  some  cases  alike,  and  in  others  unlike. 

27.  The  equation 

Becomes  when  reduced,       x=. 
That  is, 

Here  the  first  value  of  x  is  —  4-j-6z=-f-  2  I  one  positive,    and 
And  the  second  is  —  4  —  6—  —  10  )  the  other  negative. 

28.  The  equation  x2  —  8x=  —  15 

Becomes  when  reduced     a?=:4±\/16  —  15 
That  is, 


QUEST.  —  How  many  results  does  the  solution  of  a  quadratic  give  ? 
In  pure  quadratics,  iu  the  whole  value  ambiguous  ?  Is  this  the  case  in 
affected  quadratics  ? 

14* 


162  ALGEBRA.  [Sect.  X. 

Here  the  first  value  of  x  is  4+1^+5  )  both  ^.^ 
And  the  second  is  4 — lzr:-f-3  * 

That  these  two  values  of  x  are  correctly  found,  may  be 
proved  by  substituting  first  one  and  then  the  other,  for  x  itself, 
in  the  original  equation.  (Art.  161.) 

Thus  52— 8X5:^25— 40— —15 
And   32— 8X3=  9— 24=— 15. 

270.  In  the  reduction  of  an  affected  quadratic  equation, 
the  value  of  the  unknown  quantity  is  frequently  found  to  be 
imaginary. 

29.  Thus  the  equation  x2—  8x=— 20 

Becomes,  when  reduced,      a:=:4=h'\/16 — 20 
That  is,  x=4±\/—4. 

Here  the  root  of  the  negative  quantity  — 4  cannot  be  as- 
signed, (Art.  214,)  and  therefore  the  value  of  x  cannot  be 
found.  There  will  be  the  same  impossibility,  in  every  in- 
stance in  which  the  negative  part  of  the  quantities  under  the 
radical  sign  is  greater  than  the  positive  part. 

271.  When  one  of  the  values  of  the  unknown  quantity  in 
a  quadratic  equation  is  imaginary,  the  other  is  so  also.     For 
both  are  equally  affected  by  the  imaginary  root. 

Thus  in  the  example  above, 

The  first  value  of  x  is  4-+V— 4» 

And  the  second  is  4 — \/ — 4 ;  each  of  which 

contains  the  imaginary  quantity  \/ — 4. 

272.  An  equation  which  when  reduced  contains  an  ima- 
ginary root,  is  often  of  use,  to  enable  us  to  determine  whether 

QUEST. — Is  the  value  of  the  unknown  quantity  ever  imaginary  ? 
When  one  of  the  values  is  imaginary,  what  is  true  of  the  other?  Are 
equations  containing  an  imaginary  root  of  any  use  ?  What  use  ? 


Arts.  270-274.]       QUADRATIC  EQUATIONS.  163 

a  proposed  question  admits  of  an  answer,  or  involves  an  ab- 
surdity. 

30.  Suppose  it  is  required  to  divide  8  into  two  such  parts 
that  the  product  will  be  20. 

If  x  is  one  of  the  parts,  the  other  will  be  8 — x. 

By  the  conditions  proposed  (8 — z)Xz— 20 

This  becomes  when  reduced,  x=4dzVf — 4. 

Here  the  imaginary  expression  \/ — 4  shows  that  an  an- 
swer is  impossible  ;  and  that  there  is  ,an  absurdity  in  suppo- 
sing that  8  may  be  divided  into  two  such  parts,  that  their 
product  shall  be  20. 

273.  Although  a  quadratic  equation  gives  two  results,  yet 
both  these  may  not  always  be  applicable  to  the  subject  pro- 
posed.    The  quantity  under  the  radical  sign  may  be  produced 
either  from  a  positive  or   a  negative  root.     But  both  these 
roots  may  not,  in  every  instance,  belong  to  the  problem  to  be 
solved.     (Art.  251.) 

31.  Divide  the  number  30  into  two  such  parts,  that  their 
product  may  be  equal  to  8  times  their  difference. 

If  z=  the  less,  then  30 — z—  the  greater  part. 

By  the  supposition,  zX(30— x)= 8X(30— 2z). 

This  reduced,  gives  x— 23±  17=40  or  6=  the  less  part. 

But  as  40  cannot  be  a  part  of  30,  the  problem  can  have 
but  one  real  solution,  making  the  less  part  6,  and  the  greater 
part  24. 

274.  The  preceding  principles  may  be  summed  up  in  the 
following 

GENERAL     RULE. 

I.  Transpose  all  the  unknown  quantities  to  one  side  of  the 
equation  ;  and  the  known  quantities  to  the  other. 

Q,UF.ST. — Are  both  of  the  results  of  a  quadratic  always  applicable  to 
the  problem  under  consideration  ?  What  is  the  general  rule  for  the 
solution  of  quadratic  equations  ? 


164  ALGEBRA.  [Sect.  X. 

II.  Make  the  square  of  the  unknown  quantity  positive  (if  it 
is  not  already)  by  changing  the  signs  of  all  the  terms  on  both 
sides  ;  and  place  it  for  thejirst  or  leading  term.   (Art.  261.) 

III.  To  complete  the  square, 

1.  Remove  the  co  -efficient  of  the  second  power  of  the  un- 
known quantity,  and  add  the  square  of  half  of  the  co-efficient 
of  thejirst  power  of  the  unknown  quantity  to  loth  sides  of  the 
equation.     (Art.  257.)     Or, 

2.  Multiply  the  equation  by  four  times  the  co-efficient  of 
the  highest  power  of  the  unknown  quantity,  and  add  to  both 
sides  the  square  of  the  co-efficient  of  thejirst  power  of  the 
unknown  quantity.     (Art.  264.) 

IV.  Reduce  the  equation  by  extracting  the  square  root  of 
both  sides  ;  and  transpose  the  known  part  of  the  binomial 
root  thus  obtained  to  the  opposite  side.     (Art.  255.) 

-W/     EXAMPLES     FOR     PRACTICE. 

1.  Reduce  3x2  —  9x  —  4=80. 

Ol? 

2.  Reduce  4x  --  —  =46. 

x 


3.  Reduce  4x—      ll—  14. 
x+l 


4.  Reduce  5x-  =2x+ 


16       100—  9x     Q 
5.  Reduce  ---  —  -  —  =3. 
x  4x2 


6.  Reduce 


Q      „      ,  X3  — 10x2-fl 

8.  Reduce  — ^-J— 

x2 —  6x-{~9 


Art.  274.]  QUADRATIC    EQUATIONS.  165 

9.  Reduce 


10.  Reduce  -_  —          =0:—  9. 
6 


11.  Reduce  -  +  -  =  -. 

ax      a 

12.  Reduce  x*+ax2=b. 

13.  Reduce       _= 


14.  Reduce 

15.  Reduce 

16.  Reduce  2x4—x2  +96=99. 

17.  Reduce  (10+x)*— 

18.  Reduce  3x2n—2xn=8. 


19.  Reduce  2(l+x — x2) — Vl+x — x2=z — £. 

20.  Reduce 

21.  Reduce 

22.  Reduce 

23.  Reduce 

24.  Reduce 

25.  Reduce  x+16— 7-V: 

26.  Reduce 


. 


3z—  7 

27.  Reduce  ---  -  —     = 


• 
x  3x+7         13x 


166  ALGEBRA.  [Sect.  X. 

28.  Reduce  -l--"    X-  «K       - 


29.  Reduce  (z—  5)  —  3(z—  5)*=40.  ^ 

30.  Reduce  *4V^=2+3V7p. 

#**  *-  i  v  •    ,4 

PROBLEMS  IN  QUADRATIC  EQUATIONS. 

Prob.  1.  A  merchant  has  a  piece   of  cotton  cloth,  and  a 
piece  of  silk.     The  number  of  yards  in  both  is  110:  and  if 
the  square  of  the  number  of  yards  of  silk  be  subtracted  from 
80  times  the  number  of  yards  of  cotton,  the   difference  will 
be  400.     How  many  yards  are  there  in  each  piece  ? 
Let  #=  the  yards  of  silk. 
Then  110  —  z=the  yards  of  cotton. 
By  supposition     80  X  (  1  10—  a)  —  a:2=400. 
Therefore  x=  —  40±\/  10000=—  40;±=  100. 

The  first  value  of  x,  is  —  40-|-  100=60,  the  yards  of  silk  ; 
And  1  10  —  a?=  1  10  —  60=50,  the  yards  of  cotton. 

The  second  value  of  x,  is  —  40—  100=—  140  ;  but  as  this 
is  a  negative  quantity,  it  is  not  applicable  to  goods  which  a 
man  has  in  his  possession.  «  t 

Prob.  2.  The  ages  of  two  brothers  are  such,  that  their  sum  n 
is  45  years,  and  their  product  500.     What  is  the  age  of  each  ?  *" 

Prob.  3.  To  find  two  numbers  such,  that  their  difference 
shall  be  4,  and  their  product  117. 

Prob.  4.  A  merchant  having  sold  a  piece  of  cloth  which 
cost  hin^O  dollars,  found  that  if  the  price  for  which  he  sold 
it  were  multiplied  by  his  gain,  the  product  would  be  equal  to 
the  cube  of  his  gain.  What  was  his  gain  ?  !* 

Prob.  5.  To  find  two  numbers  whose  difference  shall  be  3, 
and  the  difference  of  their  cubes  117. 


Art.  274.]         QUADRATIC  EQUATIONS.  167 

Prob.  6.  To  find  two  numbers  whose  difference  shall  be 
12,  and  the  sum  of  their  squares  1424.  J),  (,) 

Prob.  7.  Two  persons  draw  prizes  in  a  lottery,  the  differ- 
ence of  which  is  120  dollars,  and  the  greater  is  to  the  less, 
as  the  less  to  10.  What  are  the  prizes  ?  *f  £>  ' 

Prob.  8.  What  two  numbers  are  those  whose  sum  is  6,  and 
the  sum  of  their  cubes  72  ?  /  / 

J  £r<~~~ 

Prob.  9.  Divide  the  number  56  into  two  such  parts,  that 
their  product  shall  be  640.  q  j  |  L 

Prob.  10.  A  gentleman  bought  a  number  of  pieces  of  cloth 
for  675  dollars,  which  he  sold  again  at  48  dollars  by  the  piece, 
and  gained  by  the  bargain  as  much  as  one  piece  cost  him. 
What  was  the  number  of  pieces  ?  J  A~~ 

Prob.  11.  A  and  B  started  together,  for  a  place  150  miles 
distant.     A's  hourly  progress  was  3  miles  more  than  B's,  and     ' .-., 
he  arrived  at  his  journey's  end  8  hours  and  20  minutes  before 
B.     What  was  the  hourly  progress  of  each  ?  $    $  ivi/rK/ 

Prob.  12.  The  difference  of  two  numbers  is  6 ;  and  if  47 
be  added  to  twice  the  square  of  the  less,  it  will  be  equal  to 
the  square  of  the  greater.  What  are  the  numbers  ?  // 

Prob.  13.  A  and  B  distributed  1200  dollars  each,  among 
a  certain  number  of  persons.     A  relieved  40  persons  more 
than  B,  and  B  gave  to  each  individual  5  dollars  more  than  A.  » 
How  many  were  relieved  by  A  and  B  ?  /jduO  ^  !    *   ^0     (*&* 

Prob.  14.  Find  two  numbers  whose  sum  is  10,  and  the  sum 
of  their  squares  58.  y  f-  ji 

Prob.  15.  Several  gentlemen  made  a  purchase  in  company 
for  175  dollars.  Two  of  them  having  withdrawn,  the  bill 
was  paid  by  the  others,  each  furnishing  10  dollars  more  than 
would  have  been  his  equal  share  if  the  bill  had  been  paid  by 
the  whole  company.  What  was  the  number  in  the  company 
at  first  ? 


168  ALGEBRA.  [Sect.  X. 

Prob.  16.  A  merchant  bought  several  yards  of  linen  for 
60  dollars,  out  of  .which  he  reserved  15  yards,  and  sold  the 
remainder  for  54  dollars,  gaining  10  cents  a  yard.  How 
many  yards  did  he  buy,  and  at  what  price  ? 

Prob.  17.  A  and  B  set  out  from  two  towns,  which  were 
247  miles  distant,  and  travelled  the  direct  road  till  they  met. 
A  went  9  miles  a  day ;  and  the  number  of  days  which  they 
travelled  before  meeting,  was  greater  by  3,  than  the  number 
of  miles  which  B  went  in  a  day.  How  many  miles  did  each 
travel  ? 

Prob.  18.  A  gentleman  bought  two  pieces  of  cloth,  the  finer 
of  which  cost  4  shillings  a  yard  more  than  the  other.  The 
finer  piece  cost  <£18  ;  but  the  coarser  one,  which  was  2  yards 
longer  than  the  finer,  cost  only  <£16.  How  many  yards  were 
there  in  each  piece,  and  what  was  the  price  of  a  yard  of  each  ? 

Prob.  19.  A  merchant  bought  54  gallons  of  Madeira  wine, 
and  a  certain  quantity  of  Teneriffe.  For  the  former,  he  gave 
half  as  many  shillings  by  the  gallon,  as  there  were  gallons 
of  TenerifFe,  and  for  the  latter,  4  shillings  less  by  the  gallon. 
He  sold  the  mixture  at  10  shillings  by  the  gallon,  and  lost 
^28  16s.  by  his  bargain.  Required  the  price  of  the  Madeira, 
and  the  number  of  gallons  of  Teneriffe. 

Prob.  20.  If  the  square  of  a  certain  number  be  taken  from 
40,  and  the  square  root  of  this  difference  be  increased  by  10, 
and  the  sum  be  multiplied  by  2,  and  the  product  divided  by 
the  number  itself,  the  quotient  will  be  4.  What  is  the  number? 

Prob.  21.  A  person  being  asked  his  age,  replied,  If  you 
add  the  square  root  of  it  to  half  of  it,  and  subtract  12,  the 
remainder  will  be  nothing.  What  was  his  age  ?  ^,  /£. 

Prob.  22.  Two  casks  of  wine  were  purchased  for  58  dol- 
lars, one  of  which  contained  5  gallons  more  than  the  other, 
and  the  price  by  the  gallon,  was  2  dollars  less  than  J  of  the 


Art.  274.]         QUADRATIC  EQUATIONS.  169 

number  of  gallons  in  the  smaller  cask.     Required  the  number 
of  gallons  in  each,  and  the  price  by  the  gallon. 

Prob.  23.  In  a  parcel  which  contains  24  coins  of  silver  and 
copper,  each  silver  coin  is  worth  as  many  cents  as  there  are 
copper  coins,  and  each  copper  coin  is  worth  as  many  cents  as 
there  are  silver  coins  ;  and  the  whole  are  worth  2  dollars  and 
16  cents.  How  many  are  there  of  each  ? 

Prob.  24.  A  person  bought  a  certain  number  of  oxen  for 
80  guineas.  If  he  had  received  4  more  oxen  for  the  same 
money,  he  would  have  paid  one  guinea  less  for  each.  What 
was  the  number  of  oxen  ? 

Prob.  25.  It  is  required  to  divide  24  into  two  such  parts 
that  their  product  shall  be  equal  to  35  times  their  difference. 

Prob.  26.  The  sum  of  two  numbers  is  60,  and  their  product 
is  to  the  sum  of  their  squares  as  2  to  5.  What  are  the  num- 
bers ? 

Prob.  27.  Divide  146  into  two  such  parts,  that  the  difference 
of  their  square  roots  may  be  6. 

Prob.  28.  What  two  numbers  are  those  whose  difference  is 
16,  and  their  product  36  ? 

Prob.  29.  Find  two  numbers  whose  sum  shall  be  1J  and 
the  sum  of  their  reciprocals  3£. 

Prob.  30.  Required  to  find  two  numbers  whose  difference 
is  15,  and  half  of  their  product  is  equal  to  the  cube  of  the  less 
number  ? 

Prob.  31.  A  company  incurred  a  bill  of  «£8  15s.  Two  of 
them  absconded  before  it  was  paid,  and  in  consequence,  those 
who  remained  had  to  pay  10s.  a  piece  more  than  their  just 
share.  How  many  were  there  in  the  company  ? 

Prob.  32.  A  gentleman  bequeathed  <£7  4s.  to  his  grandchil- 
dren ;  but  before  the  money  was  distributed  two  more  were 
15 


170  ALGEBRA.  [Sect.  X. 

added  to  their  number,  and  consequently  the  former  received 
one  shilling  a  piece  less  than  they  otherwise  would  have  done. 
How  many  grandchildren  did  he  leave  ? 

Prob.  33.  The  length  added  to  the  breadth  of  a  rectangu- 
lar room  makes  42  feet,  and  the  room  contains  432  square 
feet.  Required  the  length  and  breadth. 

Prob.  34.  A  says  to  B,  "  the  product  of  our  years  is  120  ; 
and  if  I  were  3  years  younger,  and  you  were  2  years  older, 
the  product  of  our  ages  would  still  be  120."  How  old  was 
each  ? 

Prob.  35.  Should  the  square  of  a  certain  number  be  taken 
from  89,  and  the  square  root  of  their  difference  be  increased 
by  12,  and  the  sum  multiplied  by  4,  and  the  product  divided 
by  the  number  itself,  the  quotient  will  be  16.  What  is  the 
number  ? 

Prob.  36.  A  mason  laid  105  rods  of  wall,  and  on  reflection 
found  that  if  he  had  laid  2  rods  less  per  day,  he  would  have 
been  6  days  longer  in  accomplishing  the  job.  How  many 
rods  did  he  build  per  day  ? 

Prob.  37.  The  length  of  a  gentleman's  garden  exceeded 
its  breadth  by  5  rods.  It  cost  him  3  dols.  per  rod  to  fence 
it ;  and  the  whole  number  of  dollars  was  equal  to  the  number 
of  square  rods  in  the  garden.  What  were  its  length  and 
breadth  ? 

Prob.  38.  What  number  is  that,  which  being  added  to  its 
square  root  will  make  156  ? 

Prob.  39.  The  circumference  of  a  grass-plot  is  48  yards, 
and  its  area  is  equal  to  35  times  the  difference  of  its  length 
and  breadth.  What  are  its  length  and  breadth  ? 

Prob.  40.  A  gentleman  purchased  a  building  lot,  and  in 
the  center  of  it,  erected  a  house  54  feet  long  and  36  feet  wide, 


Art.  274.]  QUADRATIC    EQUATIONS.  171 

which  covered  just  one  half  his  land.  This  arrangement  left 
him  a  flower  border  of  uniform  width  all  round  his  house. 
What  was  the  width  of  his  border,  what  the  length  and  breadth 
of  his  lot,  and  how  much  land  did  he  buy  ? 

Prob.  41.  A  general  wished  to  arrange  his  army,  which 
consisted  of  1200  men,  in  a  solid  body,  so  that  each  rank 
should  exceed  each  file  by  59  men.  How  many  must  he 
place  in  rank  and  file  ? 

Prob.  42.  A  man  has  a  painting  18  inches  long,  and  12 
inches  wide,  which  he  orders  the  cabinet  maker  to  put  into  a 
frame  of  uniform  width,  and  to  have  the  area  of  the  frame 
equal  to  that  of  the  painting.  Of  what  width  will  the  frame  be  ? 

Prob.  43.  A  and  B  together  invest  8500  in  business,  of 
which  each  put  in  a  certain  share.  A's  money  continued  in 
trade  5  months,  B's  only  two  months,  and  each  received  back 
$450  for  his  capital  and  profit.  What  share  of  the  stock 
did  each  contribute  ? 

Prob.  44.  A  merchant  sold  a  quantity  of  goods  for  <£39, 
and  gained  as  much  per  cent,  as  the  goods  cost  him.  How 
much  did  he  pay  for  the  goods  ? 

Prob.  45.  A  farmer  bought  a  flock  of  sheep  for  ,£60.  Af- 
ter selecting  15  of  the  best,  he  sold  the  remainder  for  ,£54, 
and  gained  thereby  2  shillings  a  head.  How  many  sheep 
did  he  buy,  and  what  was  the  price  of  each  ? 

Prob.  46.  A  and  B  started  from  two  cities  247  miles  apart, 
and  travelled  the  same  road  till  they  met.  A*s  progress  was 
9  miles  per  day,  and  the  number  of  days  before  they  met 
was  greater  by  3  than  the  number  of  miles  B  went  per  day. 
How  many  miles  did  each  travel  ? 

Prob.  47.  Two  persons,  A  and  B,  invest  $900  in  business. 
A's  money  remained  in  trade  4  months,  and  he  received  $512 


172  ALGEBRA.  [Sect.  XI. 

for  his  share  of  the  profit  and  stock  ;  B's  money  was  in  trade 
7  months,  and  he  received  $469  for  his  share  of  the  profit 
and  stock.  What  was  each  partner's  stock  ? 

Prob.  48.  A  merchant  bought  a  piece  of  cloth  for  $54 ; 
the  number  of  shillings  which  he  paid  per  yard  was  f  of  the 
number  of  yards.  Required  the  length  of  the  cloth,  and  the 
price  per  yard. 

Prob.  49.  There  was  a  cask  containing  20  gallons  of  wine  ; 
a  quantity  of  this  was  drawn  off  and  put  into  another  cask  of 
equal  size,  and  then  this  last  was  filled  with  water ;  and  af- 
terwards the  first  cask  was  filled  with  the  mixture  from  the 
second.  It  appears  that  if  6f  gallons  are  now  drawn  from 
the  first  and  put  into  the  second,  there  will  be  equal  quanti- 
ties of  wine  in  each  cask.  How  much  wine  was  first  drawn 
off? 

Prob.  50.  A  man  bought  80  Ibs.  of  pepper  and  100  Ibs.  of 
ginger  for  <£65,  at  such  prices  that  he  obtained  60  Ibs.  more 
of  ginger  for  <£20  than  he  did  of  pepper  for  .£10.  What 
did  he  pay  per  pound  for  each  ? 


SECTION    XI. 

TWO     UNKNOWN     QUANTITIES 


275.  IN  the  examples  given  in  the  preceding  sections,  each 
problem  has  contained  only  one  unknown  quantity.  Or  if,  in 
some  instances,  there  have  been  two,  they  have  been  so  re- 
lated to  each  other,  that  they  have  both  been  expressed  by 
means  of  the  same  letter. 


Arts.  275-277.]      UNKNOWN  QUANTITIES.  173 

But  cases  frequently  occur,  in  which  two  unknown  quanti- 
ties are  necessarily  introduced  into  the  same  calculation. 
Suppose  the  following  equations  are  given. 

1.  z+y=14 

2.  x—y—2. 

If  y  be  transposed  in  each,  they  will  become 


2.  x=2+y. 

Here  the  first  member  of  each  of  the  equations  is  z,  and 
the  second  member  of  each  is  equal  to  x.  But  according  to 
Axiom  7,  quantities  which  are  respectively  equal  to  any  other 
are  equal  to  each  other  ;  therefore, 

2+y—  14  —  y,  and  y=6. 

By  substituting  the  value  of  y  in  the  1st  equation,  (Art. 
159,)  we  have  x+6z=14  ;  then  x=S. 

276.  In  solving  the  preceding  problem,  it  will  be  observed 
that  we  first  found   the  value  of  the   unknown  quantity  #,  in 
each  equation  ;  and  then  by  making  one  of  the   expressions 
denoting  the  value  of  x  equal  to  the  other,   (Axiom  7,)  we 
formed  a  new  equation,  which  contained  only  the  other  un- 
known quantity  y. 

This  process  is  called  extermination  or  elimination. 
There  are   three   methods  of  extermination,  viz.  by  com- 
parison, by  substitution,  and  by  addition  and  subtraction. 

EXTERMINATION     BY     COMPARISON. 

277.  CASE  I.  To  exterminate  one  of  two  unknown  quan- 
tities by  comparison. 

QUEST.  —  How  are  problems  solved  which  contain  two  unknown 
quantities  ?  What  is  this  process  called  ?  How  many  methods  of  ex- 
termination ?  Name  them.  . 

15* 


174  ALGEBRA.  [Sect.  XI. 

Find  the  value  of  one  of  the  unknown  quantities  in  each  of 
the  equations,  and  form  a  new  equation  by  making  one  of 
these  values  equal  to  the  other. 

Prob.  1.  Given  x+y  =  36  )    TQ  find  ^    ^        f     and 
And     x—y  =  12  f 

1.  In  the  first  equation  a;-{-yz=:36 

2.  In  the  second  equation  x  —  y—  12 

3.  Transposing  y  in  first  equation  x=36  —  y 

4.  Transposing  y  in  second  equation  x=12-\-y 

5.  Making  3d  and  4th  equal,  (Ax.  7,)  I2-\-y—36  —  y 

6.  Transposing,  &c.  y—  12 
Substituting  value  ofy  in  4th  (Art.  159,)  z=  12+  12—  24. 
Prob.  2.  Given  2x3=28 

To  find  the  value  of  x  and 


And 
Prob.  3.  Given    4x+y=43 


Given    4x+y=43  )  ^ 

And     5x+2^56)Tofindthevalueofxand^ 
Prob.  4.  Given  4z—  2w=  16  ) 

And        6x=9y  r  To  find  the  value  of  x  and  y  • 

Prob.  5.  Given  4x  —  2«/=20 


And  To  find  the  value  of  x  and 


Prob.  6.  Given 


And  the 


Prob.  7.  To  find  two  numbers  such,  that  their  sum  shall 
be  24  ;  and  the  greater  shall  be  equal  to  five  times  the  less. 

Let  z=  the  greater  ;    and  y=  the  less. 

Prob.  8.  To  find  one  of  two  quantities,  whose  sum  is  equal 
to  h  ;  and  the  difference  of  whose  squares  is  equal  to  d. 


Prob.  9.  Given  ax+by=h  )    To  find 
And        x4-v— d ) 


QUEST.— What  is  the  rule  to  exterminate  one  of  two  unknown 
quantities  by  comparison  ? 


Arts.  278,  279.]         UNKNOWN  QUANTITIES,  175 

278.  When  the  value  of  one  of  the  unknown  quantities  is 
determined,  the  other  may  be  easily  obtained  by  substituting, 
in  one  of  the  previous  equations,  the  value  of  the  one  found 
for  the  quantity  itself.     (Art.  159.) 

The  rule  given  above,  may  be  generally  applied  for  the 
extermination  of  unknown  quantities.  But  there  are  cases  in 
which  other  methods  will  be  found  more  expeditious. 

Prob.  10.  Suppose  z=% 

And  ax-\-bx=y2 

As  in  the  first  of  these  equations  z  is  equal  to  hy,  we  may 
in  the  second  equation  substitute  this  value  of  x  instead  of  z 
itself.  The  second  equation  will  then  become,  ahy-\-bhy=y2  . 

The  equality  of  the  two  sides  is  not  affected  by  this  altera- 
tion, because  we  only  change  one  quantity  x  for  another 
which  is  equal  to  it.  By  this  means  we  obtain  an  equation 
which  contains  only  one  unknown  quantity. 

This  process  is  called  extermination  by  substitution.   Hence, 

279.  CASE  II.   To  exterminate  an  unknown   quantity  by 
substitution. 

Find  the  value  of  one  of  the  unknown  quantities,  in  one  of 
the  equations  ;  and  then  in  the  other  equation,  SUBSTITUTE 
this  value  for  the  unknown  quantity  itself.  (Art.  159.) 

Prob.  11.  Given 


find        ^ 
And  4z-|-5y=32 
Transposing  3y  in  the  1st  equation,  zrrlS  —  3y. 
Substituting  the  value  of  x  in  the  2d  equation,    (Art.  159,) 
we  have  60—  I2y+5y=32 

Then  y—± 

And  zr=15—  12=3. 

QUEST.  —  After  the  value  of  one  unknown  quantity  is  found,  how 
obtain  the  other  ?  What  is  the  second  method  of  extermination  called  ? 
What  is  the  rule  ? 


176  ALGEBBA.  [Sect.  XI. 

Prob.  12.  Given  8*+y=42  )  T   fi  d  h     rf       f    and 
And  2z+4y=18  J 

Prob.  13.  Given  2H-8y=84)Tofindthevalueofxand 
And    4z+6y=68> 

Prob.  14.  Given  3*+3y=72  1  Tofind  thevalueofxandy. 
And  4z5:rrll6  ) 


Prob.  15.  Given 
And 

Prob.  16.  A  privateer  in  chase  of  a  ship  20  miles  distant, 
sails  8  miles,  while  the  ship  sails  7.  How  far  will  each  sail 
before  the  privateer  will  overtake  the  ship  ? 

Prob.  17.  The  ages  of  two  persons,  A  and  B,  are  such  that 
seven  years  ago,  A  was  three  times  as  old  as  B  ;  and  seven 
years  hence,  A  will  be  twice  as  old  as  B.  What  is  the  age 
of  each  ? 

Prob.  18.  There  are  two  numbers,  of  which  the  greater  is 
to  the  less  as  3  to  2  ;  and  their  sum  is  the  6th  part  of  their 
product.  What  are  the  numbers  ? 

280.  There  is  a  third  method  of  exterminating  an  unknown 
quantity  from  an  equation,  which  in  many  cases,  is  preferable 
to  either  of  the  preceding. 

Prob.  19.  Suppose  that     x-\-3y=a 
And  x  —  3y—  6 

If  we  add  together  the  first  members  of  these  two  equa- 
tions, and  also  the  second  members  we  shall  have 

2x=a+b, 

an  equation  which  contains  only  the  unknown  quantity  #. 
The  other,  having  equal  co-efficients  with  contrary  signs,  has 
disappeared.  (Art.  54.)  The  equality  of  the  sides  is  pre- 
served because  we  have  only  added  equal  quantities  to  equal 
quantities. 


Arts.  280,  281.]         UNKNOWN  QUANTITIES.  177 

Again,  suppose  3x-\-y=h 

And  2x+y—  d 

If  we  subtract  the  last  equation  from  the  first,  we  shall  have 

x=h—d, 

where  y  is  exterminated,  without  affecting  the  equality  of  the 
sides. 

Again,  suppose  z  —  2y—  a 

And  a;-}-4y—  & 

Multiplying  the  1st  by  2,      2x  —  4y=2a, 

Then  adding  the  2d  and  3d,   3x=b+2a. 

This  process  is  called  extermination  by  addition  and  sub- 
traction. Hence, 

281.  CASE  III,  To  exterminate  an  unknown  quantity  by 
addition  and  subtraction. 

Multiply  or  divide  the  equations,  if  necessary  ,  in  such  a 
manner  that  the  term  which  contains  one  of  the  unknown  quan- 
tities shall  be  the  same  in  both. 

Then  subtract  one  equation  from  the  other,  if  the  signs 
of  this  unknown  quantity  are  alike,  or  add  them  together,  if 
the  signs  are  unlike. 

N.  B.  It  must  be  kept  in  mind  that  both  members  of  an 
equation  are  always  to  be  increased  or  diminished  alike,  iri 
order  to  preserve  their  equality. 

Prob.  20.  Given  2z-Hy=20  >  ^ 

J*  >  To  find  the  value  of  x  and  y. 

And     4z--5:=r28  ) 


1.  Mult,  the  1st  equation  by  2,     4z+8y=40 

2.  The  2d  equation  is  4z+5y=:28 
Subtracting  the  2d  from  the  1st,  3^=12 
Dividing,  &c.  y—  4  ;  and  z—  2. 

QUEST.  —  What  is  the  third  method  of  extermination  called  ?  What  is 
the  rule?  What  is  the  object  of  multiplying  the  equation  by  a  certain 
quantity  ?  How  do  you  know  when  to  add  and  when  to  subtract  ? 


178  ALGEBRA.  [Sect.  XI. 

Prob.  21.  Given  2x+y=l6  )  ™    ~    , ,, 

And     3z  -  %=6  I  T°  find  the  ValUG  °f  X  and  y' 
Prob.  22.  Given  4x+3^50  |  TQ  fim]  ^  yalue  of    ^ 

And  3x— %=  6  J 

Prob.  23.  Given  3z4-2y— 38  )  ™    c    , .,        ,        f        A 

y  >  To  find  the  value  of  x  and  y. 

And    5z+4y=68  J 

Prob.  24.  Given  4z— 40= — 4y  )  ™    ,.    .  , 

2*  >  To  find  the  val.  of  x  and  y . 
And    6x — 63= — 7#  f 

Prob.  25.  The  numbers  of  two  opposing  armies  are  such, 
that  the  sum  of  both  is  21110 ;  and  twice  the  number  in  the 
greater  army,  added  to  three  times  the  number  in  the  less,  is 
52219.  What  is  the  number  in  each  army  ? 

Prob.  26.  A  boy  purchased  8  lemons  and  4  oranges  for  56 
cents.  He  afterwards  bought  3  lemons  and  8  oranges  for  60 
cents.  What  did  he  pay  for  each  ? 

Prob.  27.  The  sum  of  two  numbers  is  220,  and  if  3  times 
the  less  be  taken  from  4  times  the  greater,  the  remainder  will 
be  180.  What  are  the  numbers  ? 

282.  In  the  solution  of  the  succeeding  problems,  either  of 
the  three  rules  for  exterminating  unknown  quantities  may  be 
used  at  pleasure. 

N.  B.  That  quantity  which  is  the  least  involved  should  be 
the  one  chosen  to  be  exterminated  first. 

The  pupil  will  find  it  a  useful  exercise  to  solve  each  ex- 
ample by  each  of  the  several  methods,  and  carefully  observe 
which  is  the  most  comprehensive,  and  the  best  adapted  to 
different  classes  of  problems. 

Prob.  28.  The  mast  of  a  ship  consists  of  two  parts  :  one 
third  of  the  lower  part  added  to  one  sixth  of  the  upper  part, 
is  equal  to  28  feet ;  and  five  times  the  lower  part,  diminished 
by  six  times  the  upper  part,  is  equal  to  12  feet.  What  is  the 
height  of  the  mast  ? 


Art.  282.]  UNKNOWN  QUANTITIES.  179 

Prob.  29.  To  find  a  fraction  such  that,  if  a  unit  be  added  to 
the  numerator,  the  fraction  will  be  equal  to  J  ;  but  if  a  unit 
be  added  to  the  denominator,  the  fraction  will  be  equal  to  J. 

Let  £—  the  numerator,  And  y—  the  denominator. 

1.  By  the  first  condition,  —  =£ 

By  the  second,  TT^J 

3.  Therefore  x=4,  the  numerator. 

4.  And  y— 15,  the  denominator. 
Prob.  30.  What  two  numbers  are  those,  whose  difference 

is  to  their  sum,  as  2  to  3  ;  whose  sum  is  to  their  product,  as 
3  to  5  ? 

Prob.  31.  To  find  two  numbers  such,  that  the  product  of 
their  sum  and  difference  shall  be  5,  and  the  product  of  the 
sum  of  their  squares  and  the  difference  of  their  squares  shall 
be  65. 

Prob.  32.  To  find  two  numbers  whose  difference  is  8,  and 
product  240. 

Prob.  33.  To  find  two  numbers,  whose  difference  shall  be 
12,  and  the  sum  of  their  squares  1424. 

Prob.  34.  A  certain  number  consists  of  two  digits  or 
figures,  the  sum  of  which  is  8.  If  36  be  added  to  the  num- 
ber, the  digits  will  be  inverted.  What  is  the  number  ? 

Prob.  35.  The  united  ages  of  A  and  B  amount  to  a  certain 
number  of  years  consisting  of  two  digits,  the  sum  of  which 
is  9.  If  27  years  be  subtracted  from  the  amount  of  their 
ages,  the  digits  will  be  inverted.  What  is  the  sum  of  their 
ages  ? 

Prob.  36.  A  merchant  having  mixed  a  quantity  of  brandy 
and  gin,  found,  if  he  had  put  in  6  gallons  more  of  each,  that 
the  compound  would  have  contained  7  gallons  of  brandy  for 


- 


180  ALGEBRA.  [Sect.  XL 

every  6  of  gin ;  but  if  he  had  put  in  6  gallons  less  of  each, 
the  proportions  would  have  been  as  6  to  5.  How  many  gal- 
lons did  he  mix  of  each  ? 

THREE     UNKNOWN     QUANTITIES. 

283.  In  the   preceding  examples  of  two  unknown  quanti- 
ties, it  will  be  perceived  that  the  conditions  of  each  problem 
have  furnished  two  equations  independent  of  each  other.     It 
often  becomes  necessary  to  introduce  three  or  more  unknown 
quantities  into  a  calculation.     In  such  cases,  if  the  problem 
admits  of  a  determinate  answer,  there  will  always  arise  from 
the  conditions  as  many  equations  independent  of  each  other, 
as  there  are  unknown  quantities. 

284.  Equations  are  said  to  be  independent  when  they  ex- 
press different  conditions. 

They  are  said  to  be  dependent  when  they  express  the  same 
conditions  under  different  forms.  The  former  are  not  con- 
vertible into  each  other ;  but  the  latter  may  be  changed  from 
one  form  to  the  other.  Thus  b — %—y ;  and  b=y-^-x,  are 
dependent  equations,  because  one  is  formed  from  the  other 
by  merely  transposing  x. 

Obser.  Equations  are  said  to  be  identical  when  they  express  the  same 
thing  in  the  same  form  ;  as  4x— 6  =  4r-6. 

Prob.  37.  Suppose  x+y+z=l2      ) 

4     -  ,  O  1A    f  are     SlVen     t0      filld     *1 

And        x+2y — 2%=10  > 

V  y,  and  z. 

And         x-\-y — z=4         } 

From  these  three  equations,  two  others  may  be  derived 
which  shall  contain  only  two  unknown  quantities.  One  of 
the  three  in  the  original  equations  may  be  exterminated,  in 

Q,UEST. — How  many  independent  equations  does  a  problem  of  three 
or  more  unknown  quantities  furnish  ?  What  are  independent  equa- 
tions ?  What  are  dependent  ones  ?  What  identical  ones  ? 


Arts.  283-285.]      UNKNOWN  QUANTITIES.  181 

the  same  manner  as  when  there  are  at  first  only  two,  by  the 
rules  already  given. 

In  the  equations  given  above,  if  we  transpose  y  and  z,  we 
shall  have, 

In  the  first,        xn=12  —  y  —  z. 
In  the  second,   x=  10  —  2y-f-2z. 
In  the  third,       x=:  4  —  y-\-z. 

From  these  we  may  deduce  two  new  equations,  from  which 
x  shall  be  excluded. 

By  making  the  1st  and  2d  equal,  12—  y  —  z=W  —  2y-f-2z. 

By  making  the  2d  and  3d  equal,    10  —  2y-f-2zr=4  —  y+z. 

Reducing  the  first  of  these  two,     y=3z  —  2. 

Reducing  the  second,  y—  z-[-6. 

From  these  two  equations  one  may  be  derived  containing 
only  one  unknown  quantity, 

Making  one  equal  to  the  other,         3z  —  2—  z-f-(> 

And  z~  4.     Hence, 

285.  To  solve  a  problem  containing  three  unknown  quan- 
tities, and  producing  three  independent  equations. 

First,  from  the  three  equations  deduce  two,  containing  only 
two  unknown  quantities. 

Then,  from  these  two  deduce  one,  containing  only  one  un- 
known quantity. 

For  making  these  reductions,  the  rules  already  given  are 
sufficient.  (Arts.  277,  279,  281.) 

Prob.  38.  Given  z+5y+6z=53  ^ 

2.  And     r-f  3y+3z—  30  >  To  find  a?,  y  and  ^ 

3.  And    arz—  12      ) 


From  these  three  equations  to  derive  two,  containing  only 
two  unknown  quantities, 

QUEST.  —  Rule  for  solving  problems  with  three  unknown  quantities? 
16 


182  ALGEBRA.  [Sect.  XI. 

4.  Subtract  the  2d  from  the  1st,  2y-\-3z=23. 

5.  Subtract  the  3d  from  the  2d,  2y+2z=l8. 
From  these  two,  to  derive  one, 

6.  Subtract  the  5th  from  the  4th,  z=5. 

To  find  x  and  y,  we  have  only  to  take  their  values  from 
the  third  and  fifth  equations.  (Art.  278.) 

7.  Reducing  the  fifth,  y=9 — 2=9 — 5=4. 

8.  Transposing  in  the  third,  #=12 — z — y=12 — 5— 4=3. 
Prob.  39.  Given  z+y-|-z=12      ^ 

And     z-f2y+3z=20  (TO  find  x,  y  and  z, 
And      £z4-£y-|-z=6    } 

286.  In  many  of  the  examples  in  the  preceding  sections, 
the  processes  given  might  have  been  shortened.    But  the  object 
has  been  to  illustrate  general  principles,  rather  than  to  furnish 
specimens  of   expeditious   solutions.     The  learner  will  do 
well,  as  he  passes  along,  to  exercise   his  skill  in  abridging 
the  calculations  which  are  here  given,  or  substituting  others 
in  their  stead. 

Prob.  40.  Given      z-f-y=a  \ 

And        z-j-z  =6  >    To  find  x,  y  and  z. 
And       y-j-^  — c  ' 

Prolx  41.  Three  persons,  A,  B,  and  C,  purchase  a  horse 
for  100  dollars,  but  neither  is  able  to  pay  for  the  whole. 
The  payment  would  require, 

The  whole  of  A's  money,  together  with  half  of  B's ;  or 

The  whole  of  B's,  with  one  third  of  C's ;  or 

The  whole  of  C's,  with  one  fourth  of  A's. 
How*  much  money  had  each  ? 

287.  The  learner  must  exercise  his  own  judgment,  as  to 
the  choice  of  the   quantity  to  be  first  exterminated.     It  will 


QUEST. — How  do  you  know  which  unknown  quantity  to  extermin- 
ate first  ? 


Arts.  286-288.]      UNKNOWN  QUANTITIES.  183 

generally  be  best  to  begin  with  that  which  is  most  free  from 
co-efficients,  fractions,  radical  signs,  &c. 

Prob.  42.  The  sum  of  the  distances  which  three  persons, 
A,  B,  and  C,  have  travelled,  is  62  miles  ; 
A's  distance  is  equal  to  4  times  C's,  added  to  twice  B's  ;  and 
Twice  A's  added  to  3  times  B's,  is  equal  to  17  times  C's. 

What  are  the  respective  distances  ? 
^  Prob.  43.  Given     jx+Jy+Jz—  62  ) 

And        Jz+Jy+i*=47  >  To  find  x,  y  and  z. 
SSJ 


And 

Prob.  44.  Given  xy  =  600  \ 

And  xz=  300  I    To  find  z,  y  and  z. 

And  z=20oJ 


FOUR     UNKNOWN     QUANTITIES. 

288.  The  same  method  which  is  employed  for  the  reduc- 
tion of  three  equations,  may  be  extended  to  4,  5,  or  any  num- 
ber of  equations,  containing  as  many  unknown  quantities. 

The  unknown  quantities  may  be  exterminated,  one  after 
another,  and  the  number  of  equations  may  be  reduced  by 
successive  steps  from  five  to  four,  from  four  to  three,  from 
three  to  two,  &c. 

Prob.  45.  To  find  10,  z,  y  and  z,  from 

1.  The  equation 


2.  And  x+y+u>—9      I    „ 

,,        1rt    r  F°ur  equations. 

3.  And  x+y+z=l2 

4.  And  x+w+z=10  } 

5.  Clear,  the  1st  of  frac.  y+2z-f-w=16" 

6.  Subtract  2d  from  3d,  z  —  w=r3  •  Three  equations. 

7.  Subtract  4th  from  3d,          y  —  w=2^ 

QUEST.  —  How  are  problems  solved  containing  four  or  five  unknown 
quantities  ? 


184  ALGEBRA.  [Sect.  XI 


. 


8.  Adding  5th  and  6th,  y+3z^:19  )  ^          tions< 

9.  Subtract  7th  from  6th,  —  y+z—l  ) 

10.  Adding  8th  and  9th,  42=20.     Or  z=5^ 

11.  Transposing  in  the  8th,  y—  19  —  3z=4      I  Quantities 

12.  Transposing  in  the  3d,      x=l2  —  y  —  z=^3  \    required. 

13.  Transposing  in  the  2d,  w=  9  —  x  —  y=:2  j 

Prob.  46.  Given      w+  5Q= 
And 
And 

And        z+195:=3w> 
Answer.  w= 


Prob.  47.  There  is  a  certain  number  consisting  of  two 
digits.  The  left-hand  digit  is  equal  to  three  times  the  right- 
hand  digit  ;  and  if  twelve  be  subtracted  from  the  number 
itself,  the  remainder  will  be  equal  to  the  square  of  the  left- 
hand  digit.  What  is  the  number  ? 

Prob.  48.  If  a  certain  number  be  divided  by  the  product  of 
its  two  digits,  the  quotient  will  be  2  ;  and  if  27  be  added  to  the 
number,  the  digits  will  be  inverted.  What  is  the  number? 

Prob.  49.  There  are  two  numbers,  such,  that  if  the  less  be 
taken  from  3  times  the  greater,  the  remainder  will  be  35  ; 
and  if  4  times  the  greater  be  divided  by  3  times  the  less-f-1, 
the  quotient  will  be  equal  to  the  less.  What  are  the  numbers  ? 

Prob.  50.  There  is  a  certain  fraction,  such,  that  if  3  be 
added  to  the  numerator,  the  value  of  the  fraction  will  be  J  ; 
but  if  1  be  subtracted  from  the  denominator,  the  value  will 
be  I.  What  is  the  fraction  ? 

Prob.  51.  A  gentleman  has  two  horses,  and  a  saddle  which 
is  worth  ten  guineas.  If  the  saddle  be  put  on  thejirst  horse, 
the  value  of  both  will  be  double  that  of  the  second  horse  ;  but 
if  the  saddle  be  put  on  the  second  horse,  the  value  of  both 


Art.  288.]  UNKNOWN  QUANTITIES.  185 

will  be  less  than  that  of  the  first  horse  by  13  guineas.  What 
is  the  value  of  each  horse  ? 

Prob.  52.  Divide  the  number  90  into  4  such  parts,  that  the 
first  increased  by  2,  the  second  diminished  by  2,  the  third  mul- 
tiplied by  2,  and  the  fourth  divided  by  2,  shall  all  be  equal. 

If  x,  y,  and  z,  be  three  of  the  parts,  the  fourth  will  be 
90 — x — y — z.  And  by  the  conditions,  &c. 

Prob.  53.  Find  three  numbers,  such  that  the  first  with  J 
the  sum  of  the  second  and  third  shall  be  120 ;  the  second  with 
i  the  difference  of  the  third  and  first  shall  be  70 ;  and  J  the 
sum  of  the  three  numbers  shall  be  95. 

Prob.  54.  What  two  numbers  are  those,  whose  difference, 
sum  and  product,  are  as  the  numbers  2,  3,  and  5  ? 

Prob.  55.  A  vintner  sold  at  one  time,  20  dozen  of  port 
wine,  and  30  dozen  of  sherry ;  and  for  the  whole  received 
120  guineas.  At  another  time,  he  sold  30  dozen  of  port  and 
25  dozen  of  sherry,  at  the  same  prices  as  before  ;  and  for  the 
whole  received  140  guineas.  What  was  the  price  per  dozen 
of  each  sort  of  wine  ? 

Prob.  56.  A  merchant  having  mixed  a  certain  number  of 
gallons  of  brandy  and  water,  found  that,  if  he  had  mixed  18 
gallons  more  of  each,  he  would  have  put  into  the  mixture  8 
gallons  of  brandy  for  every  7  of  water.  But  if  he  had  mixed 
18  less  of  each,  he  would  have  put  in  5  gallons  of  brandy  for 
every  4  of  water.  How  many  gallons  of  each  did  he  mix  ? 

Prob.  57.  What  fraction  is  that,  whdse  numerator  being 
doubled,  and  the  denominator  increased  'by  7,  the  value  be- 
comes §  ;  but  the  denominator  being  doubled,  and  the  nume- 
rator increased  by  2,  the  value  becomes  f  ? 

Prob.  58.  A  person  expends  30  cents  in  apples  and  pears, 
giving  a  cent  for  4  apples  and  a  cent  for  5  pears.  He  after- 
16* 


186  ALGEBRA.  [Sect.  XI. 

wards  parts  with  half  his  apples  and  one  third  of  his  pears, 
the  cost  of  which  was  13  cents.  How  many  did  he  buy  of 
each  ?  i 

289  j  If  in  the  algebraic  statement  of  the  conditions  of  a 
problem,  the  original  equations  are  more  numerous  than  the 
unknown  quantities  ;  these  equations  will  either  be  contradic- 
tory, or  one  or  more  of  them  will  be  superfluous. 


Thus  the  equations      3^=60  > 
And  4*=20  f  a 

For  by  the  first  #=20,  while  by  the  second,  3=40. 
But  if  the  latter  be  altered,  so  as  to  give  to  x  the  same  value 
as  the  former,  it  will  be  useless,  in  the  statement  of  a  prob- 
lem.    For  nothing  can  be  determined  .  from  the  one,  which 
cannot  be  from  the  other. 

Thus  of  the  equations      3x—6Q  \ 

And  ^=10  i  °ne  1S  suPerfluous- 

290.  But  if  the  number  of  independent  equations  produced 
from  the  conditions  of  a  problem,  is  less  than  the  number  of 
unknown  quantities,  the  subject  is  not  sufficiently  limited  to 
admit  of  a  definite  answer.  If  for  instance,  in  the  equation 
#-|-y—  100,  x  andy  are  required,  there  may  be  fifty  different 
answers.  The  values  of  x  and  y  may  be  either  99  and  1, 
or  98  and  2,  or  97  and  3,  &c.  For  the  sum  of  each  of  these 
pairs  of  numbers  is  equal  to  100.  But  if  there  is  a  second 
equation  which  determines  one  of  these  quantities,  the  other 
may  then  be  found  from  the  equation  already  given.  As 
i-|-y=:100,  if  £zz:46,  y  must  be  such  a  number  as  added  to 
46  will  make  100,  that  is,  it  must  be  54.  No  other  number 
will  answer  this  condition. 


QUEST. — When  the  equations  are  more  numerous  than  the  unknown 
quantities,  what  is  said  of  them  ? 


Arts.  289-292.]         UNKNOWN  QUANTITIES.  187 

291.  For  the  sake  of  abridging  the  solution  of  a  problem, 
however,  the  number  of  independent  equations  actually  put 
upon  paper  is  frequently  less,  than  the  number  of  unknown 
quantities. 

Prob.  59.  To  find  two  numbers  whose  sum  is  30,  and  the 
difference  of  their  squares  120. 

292.  In  most  cases  also,  the  solution  of  a  problem  which 
contains  many  unknown  quantities,  may  be   abridged,  by 
particular  artifices  in  substituting  a  single  letter  for  several. 

Prob.  60.  Suppose  four  numbers,  M,  z,  y  and  z,  are  re- 
quired, of  which  the  sum  of  the  three  first  is  13,  the  sum  of 
the  two  first  and  last  17,  the  sum  of  the  first  and  two  last  18, 
the  sum  of  the  three  last  21. 

Then  1.  it+x+y=lB 

2.  u+x+z—W 

3.  u+y+z=I8 

4.  *+y+z=21. 

Let  S  be  substituted  for  the  sum  of  the  four  numbers,  that 
is,  for  w+z+y+z.  (Art.  159.)  It  will  be  seen  that  of  these 
four  equations, 

The  first  contains  all  the  letters  except  z,  that  is,  S — z— 13 
The  second  contains  all  except  y1  that  is,  S — y— 17 

The  third  contains  all  except  x,  that  is,  S — x=:18 

The  fourth  contains  all  except  w,  that  is,  S — u=21. 

Adding  all  these  equations  together,  we  have 

4S — z — y — x — u— 69 

Or,       4S— (z+y+z+n)z=69.     (Art.  67.) 
But       S=i(z-\-y-\-x-\-u)  by  substitution. 
Therefore,  4S— Sz=69,  that  is,  3S=69,  and  8=23. 
Then  putting  23  for  S,  in  the  four  equations  in  which  it  is 
first  introduced,  we  have 


188  ALGEBRA.  [Sect.  XII. 

23—2=13^  fz=:23—  13—  10 

=23—  17=6 


23—  w=2l  U—  23—  21=2. 

N.  B.  Contrivances  of  this  sort  for  facilitating  the  solution 
of  particular  problems,  must  be  left  to  be  furnished  for  the 
occasion,  by  the  ingenuity  of  the  learner.  They  are  of  a 
nature  not  to  be  taught  by  a  system  of  rules. 


SECTION    XII. 

RATIO     AND     PROPORTION. 

ART.  293.  The  design  of  mathematical  investigations,  is  to 
arrive  at  the  knowledge  of  particular  quantities,  by  comparing 
them  with  other  quantities,  either  equal  to,  or  greater  or  less 
than  those  which  are  the  objects  of  inquiry.  This  end  is  most 
commonly  attained  by  means  of  a  series  of  equations  and 
proportions.  When  we  make  use  of  equations,  we  deter- 
mine the  quantity  sought,  by  discovering  its  equality  with 
some  other  quantity  or  quantities  already  known. 

We  have  frequent  occasion,  however,  to  compare  the  un- 
known quantity  with  others  which  are  not  equal  to  it,  but  either 
greater  or  less. 

294.  Unequal  quantities  may  be  compared  with  each  other 
in  two  ways. 

QUEST. — What  is  the  design  of  mathematical  investigations?  How 
is  this  end  commonly  attained  ?  In  equations  how  is  the  value  of  the 
unknown  quantity  determined  ?  In  how  many  ways  are  unequal  quan- 
tities compared  ?  What  are  they  ? 


Arts.  293-298.]  RATIO.  189 

First,  We  may  inquire  how  much  one  of  the  quantities  is 
greater  than  the  other  :  or, 

Second,  We  may  inquire  how  many  times,  one  contains 
the  other. 

295.  The  relation  which  is  found  to  exist  between  the  two 
quantities  compared,  is  called  the  ratio  of  the  two  quantities. 

RATIO  is  of  two  kinds,  arithmetical  and  geometrical.  It 
is  also  sometimes  called,  ratio  by  subtraction,  and  ratio  by 
division. 

296.  ARITHMETICAL  RATIO  is  the  DIFFERENCE  between  two 
quantities  or  sets  of  quantities.     The  quantities  themselves 
are  called  the  terms  of  the  ratio,  that  is,  the  terms  between 
which  the  ratio  exists.     Thus  2  is  the  arithmetical  ratio  of  5 
to  3.     This  is  sometimes  expressed,  by  placing  two  points 
between  the  quantities  thus,  5  •  •  3,  which  is  the  same  as  5 — 3. 
Indeed  the  term  arithmetical  ratio,  and  its  notation  by  points, 
are  almost  needless,  and  are   seldom  used.     For  the  one  is 
only  a  substitute  for  the  word  difference,  and  the  other  for 
the  sign  — . 

297.  If  both  the  terms  of  an  arithmetical  ratio  be  multiplied 
or  divided  by  the  same  quantity,  the  ratio  will  in  effect,  be 
multiplied  or  divided  by  that  quantity. 

Thus  if  a—b—r 

Then  multiply  both  sides  by  h,  (Ax.  3,)     ha — hb=hr 

And  dividing  by  h,  (Ax.  4,)  -  —  T^T* 

ft  ft  /^ 

298.  If  the  terms  of  one  arithmetical  ratio  be  added  to,  or 
subtracted  from,  the  corresponding  terms  of  another,  the  ratio 

QUEST. — What  is  ratio  ?  Of  how  many  kinds  is  it  ?  What  are  they 
called  ?  What  is  arithmetical  ratio  ?  What  are  the  quantities  them- 
selves called  ?  If  both  the  terms  are  multiplied,  or  divided,  by  the 
same  quantity,  how  is  the  ratio  affected  ?  If  the  terms  of  one  ratio  are 
added  to  the  corresponding  terras  of  another,  how  is  the  ratio  affected  ? 


190  ALGEBRA.  [Sect.  XII. 

of  their  sum  or  difference  will  be  equal  to  the  sum  or  differ- 
ence of  the  two  ratios. 

T/fc  j        .* 

.  **        >  are  the  two  ratios, 
And  d — A  J 

Then  (a+d)-(b+h)=(a-b)+(d-h).  Foreach  =  a+d-b-h. 
And  (a— d)  -  (6— h)—(a— b)  -  (d— h).  For  each  —a-d-b+h. 
Thus  the  arithmetical  ratio  of      ll-»4  is     7, 
And  the  arithmetical  ratio  of        5-~»2  is     3. 
The  ratio  of  the  sum  of  the  terms  16»«6  is  10,  which  is 
also  the  sum  of  the  ratios  7  and  3. 

The  ratio  of  the  diff.  of  the  terms  6«»2  is  4,  which  is 
also  the  difference  of  the  ratios  7  and  3. 

299.  GEOMETRICAL  RATIO  is  that  relation  between  quanti- 
ties which  is  expressed  by  the  QUOTIENT  of  the  one  divided  by 
the  other. 

Thus  the  ratio  of  8  to  4,  is  f  or  2.  For  this  is  the  quo- 
tient of  8  divided  by  4.  In  other  words,  it  shows  how  often 
4  is  contained  in  8.  So  a  I  b  expresses  the  ratio  of  a  to  b. 

300.  The  two  quantities  compared,  are  called  a  couplet. 
The  Jirst  term  is  the  antecedent ,  and  the  last,  the  consequent. 

301.  GEOMETRICAL  RATIO  is  expressed  in  two  ways. 

1.  In  the  form  of  a  fraction,  making  the   antecedent  the 
numerator,  and  the  consequent  the  denominator ;   thus  the 

ratio  of  a  to  b  is  -.     And 
b 

2.  By  placing  a  colon  between  the  quantities  compared  ; 
thus,  alb  expresses  the  ratio  of  a  to  b. 

Obser.  The  French  mathematicians  put  the  antecedent  for  the  de- 
nominator ;  and  the  consequent  for  the  numerator.  Some  American 

QUEST. — What  is  geometrical  ratio  ?  What  is  a  couplet?  The  an- 
tecedent? The  consequent?  In  how  many  ways  is  geometrical  ratio 
expressed  ?  The  first  ?  Second  ?  What  is  the  French  mode  ?  What 
are  the  comparative  advantage*  of  the  English  and  French  methods  ? 


Arts.  299-303.]  RATIO.  191 

authors  have  followed  their  example.  It  is  believed  however  that  the 
English  method,  which  is  adopted  in  the  larger  work,  is  most  in  ac- 
cordance with  reason;  while  the  French  mode  may  perhaps  have  some 
advantage  in  practice. 

302.  Of  these  three,  the  antecedent,  the  consequent,  and 
the  ratio,  any  two  being  given,  the  other  may  be  found. 

Let  az=  the  antecedent,  c=  the  consequent,  r—  the  ratio. 

By  definition  r=  -  ;  that  is,  the  ratio  is  equal  to  the  ante. 

cedent  divided  by  the  consequent. 

Multiplying  by  e,  a— cr,  that  is,  the  antecedent  is  equal  to 
the  consequent  multiplied  into  the  ratio. 

Dividing  by  r,  c=  -,  that  is,  the  consequent  ig  equal  to 

the  antecedent  divided  by  the  ratio. 

Cor.  1.  If  two  couplets  have  their  antecedents  equal,  and 
their  consequents  equal,  their  ratios  must  be  equal.  ( Euc.  7. 5.) 

Cor.  2.  If  in  two  couplets,  the  ratios  are  equal,  and  the 
antecedents  equal,  the  consequents  are  equal ;  and  if  the 
ratios  are  equal  and  the  consequents  equal,  the  antecedents 
are  equal.  (Euclid,  9.  5.) 

303.  If  the  two  quantities  compared  are  equal,  the  ratio  is 
a  unit,  or  a  ratio  of  equality.     The  ratio  of  3X6:18  is  a 
unit,  for  the  quotient  of  any  quantity  divided  by  itself  is  1. 

If  the  antecedent  of  a  couplet  is  greater  than  the  conse- 
quent, the  ratio  is  greater  than  a  unit.  For  if  a  dividend  is 
greater  than  its  divisor,  the  quotient  is  greater  than  a  unit. 

Q.UEST — When  the  antecedent  and  consequent  are  given,  how  is 
the  ratio  found  ?  When  the  consequent  and  ratio  are  given,  how  find 
the  antecedent?  When  the  antecedent  and  ratio  are  given,  how  find 
the  consequent  ?  Wljat  is  the  first  corollary  ?  The  second  ?  If  the 
two  quantities  compared  are  equal,  what  is  the  ratio  ?  If  the  antece- 
dent is  the  largest,  what  is  the  ratio  ?  What  called  ? 


192  ALGEBRA.  [Sect.  XII. 

Thus  the  ratio  of  18:6  is  3.  (Art.  103,  Cor.)  This  is  called 
a  ratio  of  greater  inequality. 

On  the  other  hand,  if  the  antecedent  is  less  than  the  con- 
sequent, the  ratio  is  less  than  a  unit,  and  is  called  a  ratio  of 
less  inequality.  Thus  the  ratio  of  2:3,  is  less  than  a  unit, 
because  the  dividend  is  less  than  the  divisor. 

305.  INVERSE  or  RECIPROCAL  ratio  is  the  ratio  of  the  re- 
ciprocals of  two  quantities.  (Art.  32.) 

Thus  the  reciprocal  ratio  of  6  to  3,  is  £  to  •£,  that  is 


The  direct  ratio  of  a  to  b  is  -,  that  is,  the  antecedent  divided 

b 

by  the  consequent. 

The  reciprocal  ratio  is  -  :  -,  or  -  -f-  r  =  -  X  7  =  -  ; 
a    b       a       b       a       1       a 

that  is,  the  consequent  b  divided  by  the  antecedent  a. 

Hence  a  reciprocal  ratio  is  expressed  by  inverting  the  frac- 
tion which  expresses  the  direct  ratio  ;  or  when  the  notation 
is  by  points,  by  inverting  the  order  of  the  terms. 

Thus  a  is  to  6,  inversely,  as  b  to  a. 

306.  COMPOUND  RATIO  is  the  ratio  of  the  PRODUCTS  of  the 
corresponding  terms  of  two  or  more  simple  ratios. 

Thus  the  ratio  of  6:3,  is  2 

And  the  ratio  of  12:4,  is  3 


The  ratio  compounded  of  these  is  72 : 12=6 

Here  the  compound  ratio  is  obtained  by  multiplying  to- 
gether the  two  antecedents,  and  also  the  two  consequents,  of 
the  simple  ratios.  Hence  it  is  equal  to  the  product  of  the 
simple  ratios. 

QUEST. — If  the  consequent  is  the  largest,  what  is  the  ratio  ?  What 
called  ?  What  is  inverse  ratio  ?  How  expressed  ?  What  is  com- 
pound ratio  ?  Does  it  differ  from  other  ratio  in  its  nature  ? 


Arts.  305-308.]  RATIO.  193 

Compound  ratio  is  not  different  in  its  nature  from  any  other 
ratio.  The  term  is  used,  to  denote  the  origin  of  the  ratio,  in 
particular  cases. 

307.  If  in  a  series  of  ratios  the  consequent  of  each  pre- 
ceding couplet,  is  the  antecedent  of  the  following  one,  the 
ratio  of  the  first  antecedent  to  the  last  consequent,  is  equal  to 
that  which  is  compounded  of  all  the  intervening  ratios.     (Euc- 
lid, 5th  B.) 

Thus,  in  the  series  of  ratios  a: b 

blc 
eld 
dlh 

the  ratio  of  alh,  is  equal  to  that  which  is  compounded  of  the 
ratios  of  «: b,  of  6:c,  of  c!rf,  of  dlh.     For  the  compound 

ratio  by  the  last  article  is  , — — =-  or  alh.     (Art.  117.) 
bcdh     h 

308.  A  particular  class  of  compound  ratios  is  produced,  by 
multiplying  a  simple  ratio  into  itself,  or  into  another  equal 
ratio.     These  are  termed  duplicate,  triplicate,  quadruplicate, 
&c.  according  to  the  number  of  multiplications. 

A  ratio  compounded  of  two  equal  ratios,  that  is,  the  square 
of  the  simple  ratio,  is  called  a  duplicate  ratio. 

One  compounded  of  three,  that  is,  the  cube  of  the  simple 
ratio,  is  called  triplicate,  &c. 

In  a  similar  manner,  the  ratio  of  the  square  roots  of  two 
quantities,  is  called  a  subduplicate  ratio ;  that  of  the  cube  roots 
a  subtriplicate  ratio,  &c. 

Thus  the  simple  ratio  of  a  to  V,  is  alb 

The  duplicate  ratio  of  a  to  b,  is  a2lb2 

QUEST. — What  is  it  equal  to  ?  When  the  consequent  of  each  pre- 
ceding couplet  is  the  antecedent  of  the  next,  what  is  the  ratio  of  the 
first  antecedent  to  the  last  consequent  equal  to  ?  What  is  a  duplicate 
ratio?  Triplicate?  Sjibduplicate  ?  Subtriplicate? 

17 


194  ALGEBRA.  [Sect.  XII. 

The  triplicate  ratio  of  a  to  b,  is  a3: 63 
The  subduplicate  ratio  of  a  to  b,  is  \Sal\/b 
The  subtriplicate  ratio  of  a  to  6,  is  %/al$/b,  &c. 

The  terms  duplicate,  triplicate,  &c.  ought  not  to  be  con- 
founded with  double,  triple,  &c. 

The  ratio  of  6  to  2  is  6:2=3 

Double  this  ratio,  that  is,  twice  the  ratio,  is  12:2=6 

Triple  the  ratio,  i.  e.  zAree  fa'mes  the  ratio,  is  18:2=9 

The  duplicate  ratio,  i.  e.  the  square  of  the  ratio,  is  62 :22=  9 
The  triplicate  ratio,  i.  e.  the  cube  of  the  ratio,  is  63 :23=27 

309.  That  quantities  may  have  a  ratio  to  each  other,  it  is 
necessary  that  they  should  be  so  far  of  the  same  nature,  as 
that  one  can  properly  be  said  to  be  either  equal  to,  or  greater, 
or  less  than  the  other.     Thus  a  foot  has  a  ratio  to  an  inch, 
for  one  is  twelve  times  as  great  as  the  other. 

310.  From  the  mode  of  expressing  geometrical  ratios  in 
the  form  of  a.  fraction,  (Art.  301,)  it  is  obvious  that  the  ratio 
of  two  quantities  is  the  same  as  the  value  of  a  fraction  whose 
numerator  and  denominator  are  equal  to  the  antecedent  and 
consequent  of  the  given  ratio.     Hence, 

311.  To  multiply,  or  divide  both  the  antecedent  and  conse- 
quent by  the  same  quantity,  does  not  alter  the  ratio.  (Art.  112.) 
To  multiply,  or  divide  the  antecedent  alone  by  any  quantity, 
multiplies  or  divides  the  ratio ;  to  multiply  the  consequent 
alone,  divides  the  ratio ;  and  to  divide  the  consequent,  multi- 
plies  the  ratio.    (Arts.  132,  135.)     That  is,  multiplying  and 
dividing  the  antecedent  or  consequent,  has  the  same  effect  on 
the  ratio,  as  a  similar  operation,  performed  on  the  numerator 
or  denominator,  has  upon  the  value  of  a  fraction. 

QUEST. — What  effect  does  it  have  on  the  ratio  to  multiply  or  divide 
both  the  antecedent  and  consequent  by  the  same  quantity?  To  multi- 
ply or  divide  the  antecedent  only  ?  The  consequent  ? 


Arts.  309-313.]  RATIO.  195 

312.  If  to  or  from  the  terms  of  any  couplet,  there  be  add- 
ed, or  subtracted,  two  other  quantities  having  the  same  ratio, 
the  sums  or  remainders  will  also  have  the  same  ratio.  (Eu- 
clid 5  and  6.  5.)  Thus  the  ratio  of  12:3  is  the  same  as 
that  of  20:5.  And  the  ratio  of  the  sum  of  the  antecedents 
12+20  to  the  sum  of  the  consequents  3+5,  is  the  same  as 
the  ratio  of  either  couplet.  That  is, 


or     „         ^ *. 

5 


So  also  the  ratio  of  the  difference  of  the  antecedents,  to  the 
difference  of  the  consequents,  is  the  same.  That  is, 

20— -12:5— 3::  12: 3  =  20:5, 

313.  If  in  several  couplets  the  ratios  are  equal,  the  sum 
of  all  the  antecedents  has  the  same  ratio  to  the  sum  of  all  the 
consequents,  which  any  one  of  the  antecedents  has  to  its  cow- 
sequent.  (Euclid  1  and  2.  5.) 

12:6=2 

Thus  the  ratio  </ 10-5=2 

6:3=2 

Therefore  the  ratio  of  (12+ 10+8+6)  :(6+5+4+3)=2. 

EXAMPLES     FOR     PRACTICE. 

| 

1.  Which  is  the   greatest,  the  ratio  of  11:9,  or  that  of 
44:35? 

2.  Which  is  the  greatest,  the  ratio  of  a+3:£«,  or  that  of 


QUEST.  —  When  you  add  or  subtract  the  terms  of  two  couplets  hav- 
ing  the  same  ratio,  what  is  the  ratio  of  their  sum  or  difference  ?  In 
several  couplets  of  equal  ratios,  what  ratio  has  the  sum  of  all  the  ante- 
cedents  to  the  sum  of  all  the  consequents  ? 


196  ALGEBRA.  [Sect.  XIL 

3.  If  the  antecedent  of  a  couplet  be  65,  and  the  ratio  13, 
what  is  the  consequent  ? 

4.  If  the  consequent  of  a  couplet  be  7,  and  the  ratio  18, 
what  is  the  antecedent  ? 

5.  What  is  the  ratio  compounded  of  the  ratios  of  3:7,  and 
2a :  56,  and  7z-f- 1 :  3y — 2  ? 

6.  What  is  the  ratio  compounded  of  x-\-y  :5,  and#-y  '.a-\-b, 
and  a-{-b:h? 

7.  If  the  ratios  of  5z+7:2z— 3,  and  a:+2:£z+3  be  com- 
pounded, will  they  produce  a  ratio  of  greater  inequality,  or 
of  less  inequality  ? 

8.  What  is  the  ratio  compounded  of  x-\-yla,  and  x — ylb, 

j   ,    %2 — V2  i 

and  6 : !Z—  ? 

a 

9.  What  is  the  ratio  compounded  of  7:5,  and  the  dupli- 
cate ratio  of  4:9,  and  the  triplicate  ratio  of  3:2  ? 

10.  What  is  the  ratio  compounded  of  3:7,  and  the  tripli- 
cate ratio  of  a?:y,  and  the  subduplicate  ratio  of  49:9  ? 

PROPORTION. 

315.  PROPORTION  is  an  equality  of  ratios.  It  is  divided 
into  two  kinds :  Arithmetical  and  Geometrical. 

Arithmetical  proportion  is  an  equality  of  arithmetical  ra- 
tios, and  geometrical  proportion  is  an  equality  of  geometrical 
ratios.  Thus  the  numbers  6,  4,  10,  8,  are  in  arithmetical 
proportion,  because  the  difference  between  6  and  4  is  the 
same  as  the  difference  between  10  and  8.  And  the  numbers 
6,  2,  12,  4,  are  in  geometrical  proportion,  because  the  quo- 
tient of  6  divided  by  2,  is  the  same  as  the  quotient  of  12 
divided  by  4. 

QUEST. — What  is  proportion  ?  Of  how  many  kinds  is  it  ?  What  is 
arithmetical  proportion  ?  Geometrical  proportion  ? 


Arts.  315-318.]  PROPORTION.  197 

316.  Care  must  be  taken  not  to  confound  proportion  with 
ratio.     This  caution  is  the  more  necessary,  as  in  common 
discourse,  the  two  terms  are  used  indiscriminately,  or  rather, 
proportion  is  used  for  both.     The  expenses  of  one  man  are 
said  to  bear  a  greater  proportion  to  his  income,  than  those  of 
another.     But  according  to  the  definition  which  has  just  been 
given,  one  proportion  is  neither  greater  nor  less  than  another. 
For  equality  does  not  admit  of  degrees.     One  ratio  may  be 
greater  or  less  than  another.     The  ratio  of  12:2  is  greater 
than  that  of  6:2,  and  less  than  that  of  20:2.     But  these  dif- 
ferences are  not  applicable  to  proportion,  when  the  term  is 
used  in  its  technical  sense.     The  loose  signification  which  is 
so  frequently  attached  to  this  word,  may  be  proper  enough  in 
familiar  language :    for  it  is  sanctioned  by  general  usage. 
But  for  scientific  purposes,  the  distinction  between  proportion 
and  ratio  should  be  clearly  drawn,  and  cautiously  observed. 

317.  Proportion  may  be  expressed,  either  by  the  common 
sign  of  equality,  or  by  four  points  between  the  two  couplets. 

Thus      /8-*6=4"2>  or  8-. 6 ::4-- 2  )  are  arithmetical 

(  a  •  •  b—  c*'d,   or  a  •  •  b : :  c  •  •  d  J    proportions. 
And       f  12:6=r8:4,  or  12 : 6 ::  8 : 4  )  are  geometrical 

t     a'.b—dlh,  or     alblldlh?     proportions. 
The  latter  is  read,  '  the  ratio  of  a  to  b  equals  the  ratio  of  c? 
to  h  ;'  or  more  concisely.  l  a  is  to  b  as  d  to  A.' 

318.  The  first  and  last  terms  are  called  the  extremes,  and 
the  other  two  the  means.     Homologous  terms  are  either  the 
two  antecedents  or  the  two  consequents.     Analogous  terms 
are  the  antecedent  and  consequent  of  the  same  couplet. 

QUEST. — What  is  the  difference  between  ratio  and  proportion  f 
In  how  many  ways  is  proportion  expressed  ?  How  is  the  latter  read  ? 
Which  are  the  extremes  ?  Which  the  means  ?  What  are  homologous 
terms  ?  What  analogous  terms  ? 

17* 


198  ALGEBRA.  [Sect.  XII. 

319.  As  the  ratios  are  equal,  it  is  manifestly  immaterial 
which  of  the  two  couplets  is  placed  first. 

If  alb:  Icld,  then  cldl  lalb.     For  if  ^=-^  then  %=^. 

b     a  do 

320.  The  number  of  terms  must  be  at  least  four.     For 
the  equality  is  between  the  ratios  of  two  couplets ;  and  each 
couplet  must  have  an  antecedent  and  a  consequent.     There 
may  be  a  proportion,  however,  among  three  quantities.     For 
one  of  the  quantities  may  be  repeated,  so  as  to  form  two 
terms.     In  this  case  the  quantity  repeated  is  called  the  mid- 
dle term,  or  a  mean  proportional  between  the  two  other  quan- 
tities, especially  if  the  proportion  is  geometrical. 

Thus  the  numbers  8,  4,  2,  are  proportional.  That  is, 
8:4: : 4:2.  Here  4  is  both  the  consequent  in  the  first  couplet, 
and  the  antecedent  in  the  last.  It  is  therefore  a  mean  pro- 
portional between  8  and  2. 

The  last  term  is  called  a  third  proportional  to  the  two  other 
quantities.  Thus  2  is  a  third  proportional  to  8  and  4. 

321.  Inverse  or  reciprocal  proportion  is  an  equality  between 
a  direct  ratio  and  a  reciprocal  ratio. 

Thus  4:2: :  J:£ ;  that  is,  4  is  to  2,  reciprocally,  as  3  to  6. 
Sometimes  also,  the  order  of  the  terms  in  one  of  the  couplets, 
is  inverted,  without  writing  them  in  the  form  of  a  fraction. 
(Art.  305.) 

Thus  4:2::  3: 6  inversely.  In  this  case,  the  first  term  is 
to  the  second,  as  the  fourth  to  the  third ;  that  is,  the  first  divi- 
ded by  the  second,  is  equal  to  the  fourth  divided  by  the  third. 

322.  When  there  is  a  series  of  quantities,  such  that  the 
ratios  of  the  first  to  the  second,  of  the  second  to  the  third,  of 


QUEST. — Which  couplet  must  be  placed  first  ?  How  many  terms  must 
there  be  ?  Can  there  be  a  proportion  with  three  quantities  ?  What  is  the 
middle  term  called  ?  The  last  term  ?  What  is  inverse  proportion  ? 


Arts.  3 19-326.]     ARITHMETICAL  PROGRESSION.  199 

the  third  to  the  fourth,  &c.  are  all  equal ;  the  quantities  are 
said  to  be  in  continued  proportion.  The  consequent  of  each 
preceding  ratio  is  then  the  antecedent  of  the  following  one. 

Continued  proportion  is  also  called  progression. 

323.  In  the  preceding  articles  of  this  section,  the  general 
properties  of  ratio  and  proportion  have  been  defined  and  illus- 
trated. It  now  remains  to  consider  the  principles  which  are 
peculiar  to  each  kind  of  proportion,  and  attend  to  their  prac- 
tical application  in  the  solution  of  problems. 


SECTION    XIII. 

ARITHMETICAL     PROPORTION     AND     PROGRESSION. 

ART.  324.  If  four  quantities  are  in  arithmetical  proportion, 
the  sum  of  the  extremes  is  equal  to  the  sum  of  the  means. 

Thus  if  a  •  •  b : :  h  •  •  ro,  then  a-\-m=b-\-h 

For  by  supposition,  a — b  =A — m 

And  transposing  —  b  and  —  m,  a-{-m=b-{-h. 

So  in  the  proportion,  12  •  •  10 : :  1 1  •  •  9,  we  have  12+9=  10+ 1 1 . 

325.  Again  if  three  quantities  are  in  arithmetical  proportion, 
the  sum  of  the  extremes  is  equal  to  double  the  mean. 

If  a  •  •  b :  I  b  •  •  c,  then,  a — b=b — c 

And  transposing  —  b  and  —  c,  a+c=26. 

326.  Quantities,  which  increase  by  a  common  difference, 
as  2,  4,  6,  8,  10 ;  or  decrease  by  a  common  difference,  as 

QUEST. — When  four  quantities  are  in  arithmetical  proportion,  what 
ia  the  sum  of  the  extremes  equal  to  ?  When  there  are  but  three  terms 
in  the  proportion,  what  is  the  sum  of  the  extremes  equal  to  ?  What  is 
continued  arithmetical  proportion  ? 


200  ALGEBRA.  [Sect.  XIII. 

15,   12,  9,  6,  3,  are  in  continued  arithmetical  proportion. 
(Art.  322.) 

Such  a  series  is  also  called  an  arithmetical  progression ; 
and  sometimes  progression  by  difference,  or  equidijferent  se- 
ries. 

327.  When  the  quantities  increase,  they  form  what  is  call- 
ed an  ascending  series,  as  3,  5,  7,  9,  11,  &c. 

When  they  decrease,  they  form  a  descending  series,  as 
11,  9,  7,  5,  &c. 

The  natural  numbers,  1,  2,  3,  4,  5,  6,  &c.  are  in  arithmet- 
ical progression  ascending. 

328.  From  the  definition  it  is  evident  that,  in  an  ascending 
series,  each  succeeding  term  is  found,  by  adding  the  common 
difference  to  the  preceding  term. 

If  the  first  term  is  3,  and  the  common  difference  2 ; 
The  series  is  3,  5,  7,  9,  11, 13,  &c. 
If  the  first  term  is  a,  and  the  common  difference  d', 
Then  a-\-d  is  the  second  term,  a-\-d-\-d=a-}-2d  the  third, 
a-^2d+d=  a+3d  the  4th,  a+3d+d=a+4d  the  5th,  &c. 

123  4  5 

And  the  series  is  a,  a-f-df,  a-}-2d,  a-}-3d,  a-\-4d,  &c. 
If  the  first  term  and  the  common  difference  are  the  same, 
the  series  becomes  more  simple.     Thus  if  a  is  the  first  term, 
and  also  the  common  difference,  and  n  the  number  of  terms, 
Then  a-^-a—^a  is  the  second  term, 

2a-\-a=3a  the  third,  &c. 
And  the  series  is  a,  2«,  3a,  4a, na. 

329.  In  a  descending  series,  each  succeeding  term  is  found, 
by  subtracting  the  common  difference  from  the  preceding  term. 

QUEST. — What  else  is  this  series  called  ?  When  the  series  increases, 
what  is  it  called  ?  When  it  decreases,  what  ?  How  is  each  successive 
term  found  in  an  ascending  series  ?  How  in  a  descending  series  ? 


Arts.  327-330.]     ARITHMETICAL  PROGRESSION.  201 

If  a  is  the  first  term,  and  d  the  common  difference,  the  se- 

123  4  5 

ries  is  a,  a — d,  a — 2d,  a — 3d,  a — 4J,  &c. 

In  this  manner,  we  may  obtain  any  term,  by  continued  ad- 
dition or  subtraction.  But  in  a  long  series,  this  process  would 
become  tedious.  There  is  a  method  much  more  expeditious. 
By  attending  to  the  series 

a,  a+d,  «+2d,  a+3d,  a+4d,  &c. 

it  will  be  seen  that  the  number  of  times  d  is  added  to  a,  is  one 
less  than  the  number  of  the  term.     Thus, 

The  second  term  is  a-\-d,  i.  e.  a  added  to  once  d ; 

The  third  is  a-\-2d,       a  added  to  twice  d ; 

The  fourth          is  a-\-Sd,       a  added  to  thrice  d,  &c. 

So  if  the  series  be  continued, 

The  50th  term  will  be  «+49rf, 

The  100th  term  a+99df. 

If  the  series  be  descending,  the  100th  term  will  be  a — 99d. 

In  the  last  term,  the  number  of  times  d  is  added  to  o,  is 
one  less  than  the  number  of  all  the  terms.     If  then 
«7z=the  common  difference,  «— the   first  term,  zzrrthe  last, 
9i=zthe  number  of  terms,  we  shall  have,  in  all  cases, 
z—a±(n— l)Xd;  that  is, 

330.  To  find  the  last  term  of  an  ascending  series. 

Add  the  product  of  the  common  difference  into  the  number 
of  terms  less  one  to  the  first  term,  and  the  sum  will  be  the 
last  term. 

If  the  series  be  descending. 

From  the  first  term  subtract  the  product  of  the  common  dif- 
ference into  the  number  of  terms  less  one,  and  the  remainder 
will  be  the  last  term. 

QUEST. — How  is  the  last  term  of  an  ascending  series  found  ?  How 
the  last  of  a.  descending  series  ? 


202  ALGEBRA.  [Sect.  XIII. 

N.  B.  Any  other  term  may  be  found  in  the  same  way. 
For  the  series  may  be  made  to  stop  at  any  term,  and  that 
may  be  considered,  for  the  time,  as  the  last. 

Thus  the  roth  term=ai;(ro — l)Xd. 

Prob.  1.  If  the  first  term  of  an  ascending  series  is  7,  the 
common  difference  3,  and  the  number  of  terms  9,  what  is  the 
last  term?  Ans.  z—a+(n—  l)d=  7+(9— 1)X3=31. 

Prob.  2.  If  the  first  term  of  a  descending  series  is  60,  the 
common  difference  5,  and  the  number  of  terms  12,  what  is 
the  last  term?  Ans.  z=a—  (n—l)d=6Q— (12—  1)X5=5. 

Prob.  3.  If  the  first  term  of  an  ascending  series  be  9,  and 
the  common  difference  4,  what  will  the  5th  term  be  ? 

Ans.  z= a-Kro— l)Xfc9+(5—  1)X4=25. 

331.  There  is  one  other  inquiry  to  be  made  concerning  a 
series  in  arithmetical  progression.  It  is  often  necessary  to 
find  the  sum  of  all  the  terms.  This  is  called  the  summation 
of  the  series.  The  most  obvious  mode  of  obtaining  the 
amount  of  the  terms,  is  to  add  them  together.  But  the  na- 
ture of  progression  will  furnish  us  with  a  more  expeditious 
method.  « 

i 

Let  us  take,  for  instance,  the  series     3,    5,    7,    9,  11, 
And  also  the  same  inverted  11,    9,    7,    5,     3, 


The  sums  of  the  terms  will  be  14,  14,  14,  14,  14. 

Take  also  the  series        a  a-\-d,     a-\-2d,  a-\-3d-> 

And  the  same  inverted   a-j-4J,  a-J-3d,  a-\-2d,  a-\-d, 


The  sums  will  be 

Hence,  it  will  be  perceived  that  the  sum  of  all  the  terms  in 
the  double  series,  is  equal  to  the  sum  of  the  extremes  repeat- 
ed as  many  times  as  there  are  terms.  Thus, 

The  sum  of  14,  14,  14,  14  and  14—14X5. 


Arts.  331-333.]     ARITHMETICAL  PROGRESSION.  203 

And  the  sum  of  the  terms  in  the  other  double  series  is 
(2«+4d)X5. 

But  this  is  twice  the  sum  of  the  terms  in  the  single  series. 
If  then  we  put 

«—  the  first  term,  nnzthe  number  of  terms, 

z=the  last,  s—  the  sum  of  the  terms, 

we  shall  have  this  equation,  s—  —  —  X»-     Hence, 

« 

332.  To  find  the  sum  of  all  the  terms  in  an  arithmetical 
progression. 

Multiply  half  the  sum  of  the  extremes  into  the  number  of 
terms,  and  the  product  will  be  the  sum  of  the  given  series. 

Prob.  4.  What  is  the  sum  of  the  natural  series  of  numbers 
1,2,  3,4,  5,  &c.  up  to  1000? 


Ans.  5=Xn=X  1000=500500. 

At  til 

333.  The  two  formulas,  z=a±(n—  l)d,  (Art.  329,)  and 
s~  --  Xw,  (Art.  331,)  contain  five  different  quantities; 

Z 

viz.  a,  the  first  term  ;  d,  the  common  difference  ;  n,  the  num- 
ber of  terms  ;  z,  the  last  term  ;  and  s,  the  sum  of  all  the  terms. 

From  these  two  formulas  others  may  be  deduced,  by  which, 
if  any  three  of  the  jive  quantities  are  given,  the  remaining  two 
may  easily  be  found.  The  most  useful  of  these  formulas 
are  the  following. 

By  the  first  formula, 

1.  The  last  term,  z=.a±(n  —  l)d,  in  which  a,  n  and  d 
are  given. 


QUEST. — How  is  the  sum  of  all  the  terms  found?  When  the  first 
term,  the  common  difference,  and  the  number  of  terms  are  given,  how 
is  the  last  term  found? 


204  ALGEBRA.  [Sect.  XIII. 

Transposing  (n — l)d, 

2.  The  first  term,  a=z±(n — l)d,  z,  n  and  d  being  given. 
Transposing  a  in  the  1st,  and  dividing  by  n — 1, 

3.  The  common  difference,  d= -,  a,  z  and  n  being  given. 

Transposing  and  dividing, 


4.  The  number  of  terms,  n— — = — \- l,a,z  and  d  being  given. 

d 

By  the  second  formula, 

a-4-z 

5.  The  sum  of  the  terms,  s=z—~-  Xn,a,z  and  n  being  given. 

Or  by  substituting  for  z  its  value, 
s—  -  — —  Xn,  in  which  a,  n  and  d  are  given. 

Reducing  the  preceding  equation, 

6.  The  first  term,  a=. £-— - — ,  s,  d  and  n  being  given. 

Zn 

7.  The  common  difference,  d—  S~  an ,  s,  a  and  n  being  given. 

n*— n 


(2a— d)2+8ds— 
8.   The  number  of  terms  n—  — -^—      —  -^ 

cf  and  s  being  given. 

QUEST. — When  the  last  term,  the  common  difference,  and  the  num- 
ber of  terms  are  given,  how  find  the  first  term  ?  When  the  first,  and 
last,  and  the  number  of  terms  are  given,  how  find  the  common  differ- 
ence ?  When  the  first  and  last  terms,  and  the  common  difference  are 
given,  how  find  the  number  of  terms  ?  When  the  first,  and  last,  and 
the  number  of  terms  are  given,  how  find  the  sum  of  all  the  terms  ? 
When  the  sum,  difference,  and  number  of  terms  are  given,  how  find 
the  first  term  ?  When  the  first  term,  the  sum,  and  the  number  of  terms 
are  given,  how  find  the  common  difference  ?  When  the  first  term,  the 
common  difference,  and  the  sum  of  the  terms  are  given,  how  find  the 
number  of  terms  ? 


Arts.  334,  335.]     ARITHMETICAL  PROGRESSION.  205 

Obser.  A  variety  of  other  formulas  may  be  deduced  from  the  pre- 
ceding equations,  the  investigation  of  which  will  afford  the  student  a 
a  pleasing  and  profitable  exercise. 

334.  By  the  third  formula,  e.  g.  may  be  found  any  num- 
ber of  arithmetical  means,  between  two  given  numbers.  For 
the  whole  number  of  terms  consists  of  the  two  extremes  and 
all  the  intermediate  terms.  If  then  m—  number  of  means, 
w+2z=:n,  the  whole  number  of  terms.  Substituting  m+2  for 
n  in  the  third  equation,  we  have, 


The  common  difference,  c?—  —  —  ,  in  which  «,  z  and  m 

fw-j—  1 

are  given. 

Prob.  5.  Find  6  arithmetical  means,  between   1  and  43. 
.          C  The  common  difference  is  6. 

I  The  series  1,  7,  13,  19,  25,  31,  37,  43. 

335.  It  is  obvious  from  the  illustration  in  Art.  331,  that  the 
sum  of  the  extremes  in  an  arithmetical  progression,  is  equal  to 
the  sum  of  any  other  two  terms  equally  distant  from  the  ex- 
tremes. Thus,  in  the  series  3,  5,  7,  9,  11,  the  sum  of  the 
first  and  last  terms,  of  the  first  but  one  and  last  but  one,  &c., 
is  the  same  in  each  case,  viz.  14.  The  same  is  true  of  every 
series. 

Prob.  6.  If  the  first  term  of  an  increasing  arithmetical  se- 
ries is  3,  the  common  difference  2,  and  the  number  of  terms 
20  ;  what  is  the  sum  of  the  series  ? 

Prob.  7.  If  100  stones  be  placed  in  a  straight  line,  at  the 
distance  of  a  yard  from  each  other  ;  how  far  must  a  person 
travel,  to  bring  them  one  by  one  to  a  box  placed  at  the  dis- 
tance of  a  yard  from  the  first  stone  ? 

QUEST.  —  How  find  any  number  of  arithmetical  means  between  two 
given  numbers  ?  In  a  series  of  arithmetical  progression,  what  is  the 
sum  of  the  extremes  equal  to  ? 

18 


206  ALGEBRA.  [Sect.  XIII. 

Prob.  8.  What  is  the  sum  of  150  terms  of  the  series 

3'  3'  *'  ?  3'  2'  3'  &C'  ' 

Prob.  9.  If  the  sum  of  an  arithmetical  series  is  1455,  the 
least  term  5,  and  the  number  of  terms  30  ;  what  is  the  com- 
mon difference  ? 

Prob.  10.  If  the  sum  of  an  arithmetical  series  is  567,  the 
first  term  7,  and  the  common  difference  2  ;  what  is  the  num- 
ber of  terms  ? 

Prob.  11.  What  is  the  sum  of  32  terms  of  the  series 
1,  1J,  2,  2J,  3,  &c.  ? 

Prob.  12.  A  gentleman  bought  47  books,  and  gave  10  cents 
for  the  first,  30  cents  for  the  second,  50  cents  for  the  third, 

&c.     What  did  he  give  for  the  whole  ? 
• 
Prob.  13.  A  person  put  into  a  charity  box,  a  cent  the  first 

day  of  the  year,  two  cents  the  second  day,  three  cents  the 
third  day,  &c.  to  the  end  of  the  year.  What  was  the  whole 
sum  for  365  days  ? 

Prob.  14.  How  many  strokes  does  a  common  clock  strike 
in  24  hours  ? 

Prob.  15.  The  clocks  of  Venice  go  on  to  24  o'clock  ;  how 
many  strokes  do  they  strike  in  a  day  ? 

Prob.  16.  Required  the  sum  of  the  odd  numbers  1,  3,  5,  7, 
9,  &c.  continued  to  101  terms. 

Prob.  17.  Required  the  365th  term  of  the  series  of  even 
numbers  2,  4,  6,  8,  10,  12,  &c. 

Prob.  18.  The  first  term  of  a  series  is  4,  the  common  differ- 
ence 3,  and  the  number  of  terms  100 ;  what  is  the  last  term  ? 

Prob.  19.  A  man  puts  $1  at  interest  at  6  per  cent.  What 
will  be  the  amount  in  40  years  ? 


Art.  336.]  ARITHMETICAL    PROGRESSION.  207 

Prob.  20.  The  extremes  of  a  series  are  2  and  29  ;  and  the 
number  of  terms  is  ten.  What  is  the  common  difference  ? 

Prob.  21.  The  extremes  of  a  series  are  3  and  39,  and  the 
common  difference  2.     What  is  the  number  of  terms  ? 
Prob.  22.  Find  5  means  between  6  and  48. 
Prob.  23.  Find  6  means  between  8  and  36. 
.    336.  Problems  of  various  kinds,  in  arithmetical  progression, 
may  be  solved  by  stating  the  conditions  algebraically,  and 
then  reducing  the  equations. 

Prob.  24.  Find  four  numbers  in  arithmetical  progression, 
whose  sum  shall  be  56,  and  the  sum  of  their  squares  864. 
If  #—  the  second  of  the  four  numbers, 
And  y=  their  common  difference  : 
The  series  will  be  x — y,  a?,  #-f-y,  x-\-2y. 
By  the  conditions,     (x — y)-\-x-}-(x-\-y)-{-(x-\-2y)=56 
And  (x-y)2+ 

That  is, 

And  4x2 -\-4xy +6y2  = 

Reducing  these  equations,  we  have      x=12,  and  y— 4. 
The  numbers  required,  therefore,  are   8,  12,  16,  and  20. 

Prob.  25.  The  sum  of  three  numbers  in  arithmetical  pro- 
gression is  9,  and  the  sum  of  their  cubes  is  153.  What  are 
the  numbers  ? 

Prob.  26.  The  sum  of  three  numbers  in  arithmetical  pro- 
gression is  15,  and  the  sum  of  the  squares  of  the  two  ex- 
tremes is  58.  What  are  the  numbers  ? 

Prob.  27.  There  are  four  numbers  in  arithmetical  progres- 
sion :  the  sum  of  the  squares  of  the  two  first  is  34 ;  and  the 
sum  of  the  squares  of  the  two  last  is  130.  What  are  the 
numbers  ? 


208  ALGEBRA.  [Sect.  XIV. 

Prob.  28.  A  certain  number  consists  of  three  digits,  which 
are  in  arithmetical  progression,  and  the  number  divided  by 
the  sum  of  its  digits  is  equal  to  26 ;  but  if  198  be  added  to 
it,  the  digits  will  be  inverted.  What  is  the  number  ? 

Let  the  digits  be  equal  to  x — y,  #,  and  z+y,  respectively. 
Then  the  number=lW(x-y)+Wx+(x+y)=lllx-99y,  &c. 

Prob.  29.  The  sum  of  the  squares  of  the  extremes  of  four 
numbers  in  arithmetical  progression  is  200 ;  and  the  sum  of 
the  squares  of  the  means  is  136.  What  are  the  numbers  ? 

Prob.  30.  There  are  four  numbers  in  arithmetical  progres- 
sion, whose  sum  is  28,  and  their  continued  product  is  585  ? 
What  are  the  numbers  ? 


SECTION   XIV. 

GEOMETRICAL     PROPORTION    AND     PROGRESSION. 

ART.  337.  If  four  quantities  are  in  geometrical  proportion, 
the  product  of  the  extremes  is  equal  to  the  product  of  the 
means.  Thus, 

12:8::15:10;  therefore  12X10=8 X 15.     Hence, 

338.  Any  factor  may  be  transferred  from  one  of  the  means 
to  the  other,  or  from  one  extreme  to  the  other,  without  affect- 
ing the  proportion. 

Thus  if  almb:  Ixly,  then  alb:  Imxly  ;  for  the  product  of 
the  means  in  both  cases  is  the  same. 

So  if  nalbllxly,   then   albllxlny. 

QUEST. — If  four  quantities  are  in  geometrical  proportion,  what  is 
the  product  of  the  extremes  equal  to  ? 


Arts.  337-341.]     GEOMETRICAL  PROGRESSION.  209 

339.  On  the  other  hand,  if  the  product  of  two  quantities 
is  equal  to  the  product  of  two  others,  the  fpur  quantities  will 
form  a  proportion  if  they  are  so  arranged,  that  those  on  one 
side  of  the  equation  shall  constitute  the  means,  and  those  on 
the  other  side  the  extremes.     Thus  since  6X  12=8x9,  then 
6:8::9:12. 

Cor.  The  same  must  be  true  of  any  factors  which  form 
the  two  sides  of  an  equation.  Thus  if 

(a-\-b)Xc=(d — w)Xy,  then  a-\-b:d — mllylc. 

340.  If  three  quantities  are  proportional,  the  product  of  the 
extremes  is  equal  to  the  square  of  the  mean.     For  this  mean 
proportional  is,  at  the  same  time,  the  consequent  of  the  first 
couplet,  and  the  antecedent  of  the  last.     (Art.  320.)     It  is 
therefore  to  be  multiplied  into  itself,  that  is,  it  is  to  be  squared. 

Thus,  4:6:  :6:9;  therefore  4  X9z=6X  6. 
If  albl'.b'.c,  then  mult,  extremes  and  means,  ae~ ft2. 
Hence,  a  mean  proportional  between  two  quantities  may 
be  found,  by  extracting  the  square  root  of  their  product. 
If  alx:  :xic,  then  x2=ac,  and  z=\/ac.     (Art.  249.) 

341.  It  follows,  from  Art.  338,  that  in  a  proportion,  either 
extreme  is  equal  to  the  product  of  the  means,  divided  by  the 
other  extreme  ;  and  either  of  the  means  is  equal  to  the  pro- 
duct of  the  extremes,  divided  by  the  other  mean. 

1.  If  albllcld,  then  ad—lc. 

2.  Dividing  by  cZ,  a=bc-^-d. 

3.  Dividing  the  first  by  c,  b=ad-r-c. 

4.  Dividing  it  by  6,  c~ad-±-b. 

5.  Dividing  it  by  a,  d=bc-~a. 

QUEST. — How  is  an  equation  put  into  a  proportion  ?  If  three  quan- 
tities are  in  proportion,  what  is  the  product  of  extremes  equal  to? 
How  is  a  mean  proportional  between  two  quantities  found  ?  When 
the  means  and  one  extreme  are  given,  how  find  the  other  extreme  ? 
When  the  extremes  and  one  of  the  means  are  given,  how  find  the  other? 

18* 


210  ALGEBRA.  [Sect.  XIV. 

That  is,  the  fourth  term  is  equal  to  the  product  of  the  se- 
cond and  third  divided  by  the  first. 

N.  B.  On  this  principle  is  founded  the  rule  of  simple  pro- 
portion in  arithmetic,  commonly  called  the  "  Rule  of  Three" 
Three  numbers  are  given  to  find  a  fourth,  which  is  obtained 
by  multiplying  together  the  second  and  third,  and  dividing 
by  the  first. 

342.  The  propositions  respecting  the  products  of  the  means 
and  of  the  extremes,  furnish  a  very  simple  and  convenient 
criterion  for  determining  whether  any  four  quantities  are  pro- 
portional.    We  have  only  to  multiply  the  means  together, 
and  also  the  extremes.     If  the  products  are  equal,  the  quan- 
tities are  proportional.     If  the  products  are  not  equal,  the 
quantities  are  not  proportional. 

343.  It  is  evident  that  the  terms  of  a  proportion  may  un- 
dergo any  change  which  will  not  destroy  the  equality  of  the 
ratios ;  or  which  will  leave  the  product  of  the  means  equal 
to  the  product  of  the  extremes.     These  changes  are  numerous, 
but  they  may  be  reduced  to  a  few  general  principles. 

CASE  I.  Changes  in  the  order  of  the  terms. 

344.  If  four  quantities  are  proportional,  the  order  of  the 
means,  or  of  the  extremes,  or  of  the  terms  of  both  couplets, 
may  be  inverted  without  destroying  the  proportion. 

Thus  if  alb:  Icld,  and  12:8:  :6:4,  then, 

,    r  J7  *  (    al  cl  ibid  )  the  1st  is  to  the  3d  as 

1.  Inverting  the  means,*  < 

\  12:6:  :8:4  )  the  2d  to  the  4th. 

QUEST. — What  rule  is  founded  on  this  principle?  How  can  you 
tell  whether  four  quantities  are  proportional  ?  What  alterations  can 
be  made  on  the  terms  of  a  proportion  ?  When  the  means  are  invert- 
ed, what  is  it  called  ?  When  the  terms  of  each  couplet  are  inverted, 
what.?  If  the  terms  of  only  one  couplet  are  inverted,  what  is  the  ef- 
fect on  the  proportion  ? 

*  This  is  called  alternation,     (Euclid  16.  5.) 


Arts.  342-345.]     GEOMETRICAL  PROGRESSION.  211 


2.  Inverting  the  extremes,  \     '    '  * c ' 

(  4:8::6: 


12  i  as  the  3d  to  the  1st. 

3.  Inverting  the  terms  of  (  bl   a  I  ldi2  \  the  2d  is  to  the  1st  as 
each  couplet*  I  8: 12:  :4:6  J  the  4th  to  the  3d. 

4.  We  may  change  the  order  of  the  two  couplets.    (Art. 
319.) 

Cor.  The  order  of  the  whole  proportion  may  be  inverted. 
N.  B.  If  the  terms  of  only  one  of  the  couplets  are  inverted, 
the  proportion  becomes  reciprocal.     (Art.  321.) 
If  alb:  I  eld,  then  a  is  to  6,  reciprocally,  as  d  to  c. 

CASE  II.  Multiplying  or  dividing  by  the  same  quantity. 

345.  If  four  quantities  are  proportional,  the  two  analogous 
or  two  homologous  terms  may  be  multiplied  or  divided  by  the 
same  quantity,  without  destroying  the  proportion.  Thus, 

If  alb :  I  eld,  then,  if  analogous  terms  are  multiplied,  or 
divided,  the  ratios  will  not  be  altered.  (Art.  311.) 

1.  malmbllcld.  2.  albllmclmd. 

0    a   b  .      c  d 

3.   -l-llcld.  4.  albll-l-. 

mm  mm 

If  homologous  terms  be  multiplied  or  divided,  both  ratios 
will  be  equally  increased  or  diminished. 

5.  malhllmcld.  6.  almbllclmd. 

a  ,     c  b        d 

7.  ~lbll-ld.  8.  al-llcl-. 

mm  mm 

Cor.  All  the  terms  may  be  multiplied,  or  divided  by  the 

rnu  T  -abed 

same  quantity,     ihus,  malmbl  Imclmd,  or  —:— ::—:—. 

mm     mm 

Q.UEST. — If  two  analogous  terms  are  multiplied  or  divided  by  the 
same  quantity,  what  is  the  effect  ?  If  two  homologous  terms  are  mul- 
tiplied or  divided,  what  ? 

*  This  is  technically  called  inversion. 


212  ALGEBRA.  [Sect.  XIV, 

CASE  III.   Comparing  one  proportion  with  another. 

346.  If  two  ratios  are  respectively  equal  to  a  third,  they 
are  equal  to  each  other.     (Euclid  11.  5.)  ' 

This  is  nothing  more  than  the  7th  axiom  applied  to  ratios. 

'    ,  [  then  a'.bllc'.d,  or  alcllbld.     (Art.  344.) 

And  clal  Imln  " 

2.  If  albllmln)   , 

And  mini  Iclb*  then  albl  lcldl  °r  alcl  lbld' 

Cor.  If  alb:  ^nln  )  ^  a.b>c.d     (Eudid  13  5  } 

fft  •  Tt^^C  %  CL  ' 

For  if  the  ratio  of  mln  is  greater  than  that  of  eld,  it  is 
manifest  that  the  ratio  of  alb,  which  is  equal  to  that  of  01:91, 
is  also  greater  than  that  of  eld. 

346.a.  In  these  instances,  the  terms  which  are  alike  in  the 
two  proportions  are  the  twojirst  and  the  two  last,  and  the  re- 
sulting proportion  is  uniformly  direct.  But  this  arrangement 
is  not  essential.  The  order  of  the  terms  may  be  changed,  in 
various  ways,  without  affecting  the  equality  of  the  ratios. 
(Art.  344.) 

The  proposition  to  which  the  instances  of  equality  belong, 
is  usually  cited  by- the  words,  "  ex  aquo"  or  "  ex  cequali." 
(Euclid  22.  5.) 

347.  Any  number  of  proportions  may  be  compared,  in  the 
same  manner,  if  the  two  first  or  the  two  last  terms  in  each 
preceding  proportion,  are  the  same  with  the  two  first  or  the 
two  last  in  the  following  one. 

Thus  if  albllcld^ 

And        c'.dllhll  I    , 

A    ,        ,    7  >then  alb:  Ixly. 

And       hllllmln  I 

And      mini  Ixly  J 

QUEST. — When  two  ratios  are  each  equal  to  a  third,  how  are  they  to 
each  other  ?  How  is  this  proposition  cited  in  geometry  ?  When  may 
any  number  of  proportions  be  compared  in  this  manner  ? 


Arts.  346-349.]     GEOMETRICAL  PROGRESSION.  213 

That  is,  the  two  first  terms  of  the  first  proportion  have  the 
same  ratio,  as  the  two  last  terms  of  the  last  proportion.  For 
it  is  manifest  that  the  ratio  of  all  the  couplets  is  the  same. 

348.  But  if  the  two  means,  or  the  two  extremes,  in  one 
proportion,  be  the  same  with  the  means,  or  the  extremes,  in 
another,  the  four  remaining  terms  will  be  reciprocally  pro- 
portional. 

If       almllnlb).  1   1 

A     ,  _  ?  then  ale::-:-,  or  alclldlb. 

And  clmllnld  J  b  d 

For  ab=mn  )  ^Art  33?  j    Therefore  a&=crf,  and  ale:  :d:b. 
And  cd=mn  * 

In  this  example,  the  two  means  in  one  proportion,  are  like 
those  in  the  other.  But  the  principle  will  be  the  same,  if  the 
extremes  are  alike,  or  if  the  extremes  in  one  proportion  are 
like  the  means  in  the  other. 


If      mlallbln  > 

A    i  .7.     rihen  alclldlb. 

And  »*:c:  :alw  > 

Orif  «:m::n:6  )   , 

Anri    •   ••,/•*,  ctnen  «zc::rf:6. 

Ana  ml  c.  lalw  > 


The  proposition  in  geometry  which  applies  to  this  case,  is 
usually  cited  by  the  words  "  ex  aquo  perturbate"  (Euc.  23. 5.) 

CASE  IV.  Addition  and  Subtraction  of  equal  ratios. 

349.  If  to  or  from  two  analogous  or  two  homologous  terms 
of  a  proportion,  two  other  quantities  having  the  same  ratio 
be  added  or  subtracted,  the  proportion  ivill  be  preserved. 
(Euclid  2.  5.) 

QUEST. — If  the  two  means  or  two  extremes  in  one  proportion  be  the 
same  as  the  means  or  extremes  in  another,  how  are  the  remaining 
terms  ?  How  is  this  proposition  cited  in  geometry  ?  When  two  anal- 
ogous or  homologous  terms  are  added  to  or  subtracted  from  two  other 
quantities  having  the  same  ratio,  how  is  the  proportion  ? 


214  ALGEBRA.  [Sect.  XIV. 

For  a  ratio  is  not  altered,  by  adding  to  it,  or  subtracting 
from  it,  the  terms  of  another  equal  ratio.     (Art.  313.) 

If  albllcld,     and  albumin, 

Then  by  adding  to,  or  subtracting  from  a  and  b,  the  terms 
of  the  equal  ratio  mln,  we  have 

a-\-mlb-\-nl  Icld,     and  a  —  mlb  —  nllcld. 
And  by  adding  and  subtracting  m  and  n,  to  and  from  c  and 
d,  we  have 

albl  :c+w:e?+n,         and  albllc  —  mid  —  71. 
Here  the  addition  and  subtraction  are  to  and  from  analo- 
gous terms.     But  by  alternation,  (Art.  344,)  these  terms  will 
become  homologous,  and  we  shall  have 

a-{-mlcl  lb-^nld,         and  a  —  mlcllb  —  nld. 
Cor.  1.  This  addition  may,  evidently,  be  extended  to  any 
number  of  equal  ratios.     (Euclid  2.  5.  cor.) 

fc:<n 

Thus  if  albl  :  J  ™    Ithen  albl  lc+h+m+xld+l+n+y 
[x:y 

Cor.  2.  If  a:l::c:d 


.:c+n:dt  (Eu  24  5) 

Anamlbllnld) 

For  by  alternation  al  c  1  1  b  I  d  >  ^       <       a-\-m  I  c-\-n  I  ibid 
And  mlnllbld)  £  or  a-\-mlbllc-{-nld. 

350.  Hence,  if  two  analogous  or  homologous  terms  be  add- 
ed to,  or  subtracted  from  the  two  others,  the  proportion  will 
be  preserved. 

Thus,  if  albllcld,  and  12:4:  :6;2,  then, 
1.  Adding  the  two  last  terms,  to  the  two  first. 

a+clb+dllalb  12+6:   4+2::  12:4 

zuda+clb+dllcld  12+6:   4+2::   6:2 

or     a+claiib+dlb  12+6:12:  :4+  2:4 

12+6:  6::4+2:2. 


Arts.  350,  351.]     GEOMETRICAL  PROGRESSION.  215 

2.  Adding  the  two  antecedents  to  the  two  consequents. 

a+b:b::c+d:d          12+4:  4::6-f2:2 

a+blall c+dlc,  &c.     12+4:12: : 6+2: 6,  &c. 
This  is  called  composition.     (Euclid  18.  5.) 

3.  Subtracting  the  two  first  terms  from  the  two  last, 
c — alalld — 6:6,    or  c — alclld — bid,  &c. 

4.  Subtracting  the  two  last  terms  from  the  iwojirst. 
a — clb — r/::«:6,     or  a — clb — dllcld,  &c. 

5.  Subtracting  the  consequents  from  the  antecedents, 
a — 6:6:  :c — did,     or  ala — bllclc— d,  &c. 

The  alteration  expressed  by  the  last  of  these  forms  is  call- 
ed conversion. 

6.  Subtracting  the  antecedents  from  the  consequents, 
b — alalld — cic,     or  6:6 — alldld — c,  &c. 

7.  Adding  and  subtracting,     a-\-bla — bllc-{-dlc — d. 
That  is,  the  sum  of  the  two  first  terms,  is  to  their  differ- 
ence, as  the  sum  of  the  two  last,  to  their  difference. 

Cor.  If  any  compound  quantities,  arranged  as  in  the  pre- 
ceding examples,  are  proportional,  the  simple  quantities  of 
which  they  are  compounded  are  proportional  also. 

Thus,  if  «-(-6:6:  lc-}-dld,  then  alb  1 1  eld.  This  is  called 
division.  (Euclid  17.  5.) 

CASE  V.   Compounding  Proportions. 

351.  If  the  corresponding  terms  of  two  or  more  ranks  of 
proportional  quantities  be  multiplied  together,  the  product 
will  be  proportional. 

QUEST. — What  is  composition  ?  Conversion  ?  Division  ?  If  the 
corresponding  terms  of  two  or  more  ranks  of  proportionals  are  multi- 
plied together,  how  will  the  product  be  ? 


216  ALGEBRA.  [Sect.  XIV. 

This  is  compounding  ratios,  (Art.  306,)  or  compounding 
proportions.  It  should  be  distinguished  from  what  is  called 
composition,  which  is  an  addition  of  the  terms  of  a  ratio. 
(Arts.  350,  2.) 

If       aibucid  12:4::6:2 

And    hllllmln  10:5:  :8:4 


Then  ahlbl:  Icmldn  120:20:  :48:8. 

For  from  the  nature  of  proportion,  the  two  ratios  in  the 
first  rank  are  equal,  and  also  the  ratios  in  the  second  rank. 
And  multiplying  the  corresponding  terms  is  multiplying  the 
ratios,  (Art.  311,)  that  is,  multiplying  equals  by  equals, 
(Ax.  3 ;)  so  that  the  ratios  will  still  be  equal,  and  therefore 
the  four  products  must  be  proportional. 

The  same  proof  is  applicable  to  any  number  of  proportions. 

fa:&::c:<T 

If      <h:l::m:n      ihenahp'.blqllcmxldny. 

[p:q::x:yj 

From  this  it  is  evident  that  if  the  terms  of  a  proportion  be 
multiplied,  each  into  itself,  that  is,  if  they  be  raised  to  any 
power,  they  will  still  be  proportional.  (Art.  308.) 

If  a:b::c:d  2:4::6:12 

a:b::c:d  2:4::6:12 


Theua2:b2::c2:d2  4:16::36:144 

Proportionals  will  also  be  obtained,  by  reversing  this  pro- 
cess, that  is,  by  extracting  the  roots  of  the  terms. 

If  alb:  icld,         then  \/al\/b:  :\Scl\/d. 
For  taking  the  product  of  extremes  and  means,  ad=bc 
And  extracting  the  root  of  both  sides,  \^ad=z\^b  c 

That  is,  (Arts.  210.a,  339,)  \^a:^b:  :\/cl\/d. 

__ . 

QUEST. — What  is  meant  by  compounding  proportions  ?  What  is  the 
difference  between  compounding  proportions  and  composition  ?  If  sev- 
eral quantities  are  proportional,  how  are  like  powers  or  roots  of  them  ? 


Arts.  352-354.]     GEOMETRICAL  PROPORTION.  217 

CASE  VI.  Involution  and  Evolution  of  the  terms. 

352.  If  several  quantities  are  proportional,  their  like  pow- 
ers or  like  roots  are  proportional. 

If  alb:  Icld, 
Thenan:&n::cn:<F,         and  ^/al^bl  l%/cl%/d. 

mm          mm 

And  %/anl%/bnli%/cnl%/dn,  that  is,  a"  ;&":  lc"ld". 

Obser..  It  must  not  be  inferred  from  this,  that  quantities  have  the 
same  ratio  as  their  like  powers  or  like  roots.  (Art.  308.) 

353.  If  the  terms  in  one  rank  of  proportionals  be  divided 
by  the  corresponding  terms  in  another  rank,  the  quotients 
will  be  proportional. 

This  is  sometimes  called  the  resolution  of  ratios. 

If       aibiicid  12:6::18:9 

And     hllllmin  6:2::  9:3 

aft^rf  126189 

"  h:r:m:n  6:2::9:3' 

This  is  merely  reversing  the  process  in  Art.  351,  and  may 
be  demonstrated  in  a  similar  manner. 

This  should  be  distinguished  from  what  geometers  call 
division,  which  is  a  subtraction  of  the  terms  of  a  ratio.  (Art. 
350,7.) 

354.  When  proportions  are  compounded  by  multiplication, 
it  will  often  be  the  case  that  the  same  factor  will  be  found  in 
two  analogous  or  two  homologous  terms. 

Thus  if  albllcld 
And        mlallnlc 


am  lab  1 1  en  led. 

Here  a  is  in  the  two  first  terms,  and  c  in  the  two  last.     Di- 
viding by  these,  (Art.  345,)  the  proportion  becomes 
mlbllnld.     Hence, 

QUEST. — What  is  meant  by  the  resolution  of  ratios? 

19 


218  ALGEBRA.  [Sect.  XIV. 

355.  In  compounding  proportions,  equal  factors  or  divisors 
in  two  analogous  or  homologous  terms,  may  be  rejected. 

(a:b::c:d  12:4::9:3 

If  )  b:h::d:l  4:8::3:6 

f  h:mi:l:n  8:20::6:15 


Then        almllcln  12:20:  :9:15 

This  rule  may  be  applied  to  the  cases,  to  which  the  terms 
"  ex  aquo"  and  "  ex  aquo  perturbate"  refer.  (Arts.  346.0, 
348.)  One  of  the  methods  may  serve  to  verify  the  other. 

356.  When  four  quantities  are  proportional,  if  the  first  be 
greater  than  the  second,  the  third  will  be  greater  than  the 
fourth  ;  if  equal,  equal  :  if  less,  less. 


Let  albllcld;  then  if 


357.  If  four  quantities  are  proportional,  their  reciprocals 
are  proportional ;  and  v.  v. 

If  a:b::c:d,  then  -:-::-:-. 

a    b      c     d 

For  in  each  of  these  proportions,  we  have,  by  reduction, 
ad=bc. 


PROBLEMS  IN  GEOMETRICAL  PROPORTION. 

Prob.  1.  Divide  the  number  49  into  two  such  parts,  that 
the  greater  increased  by  6,  may  be  to  the  less  diminished  by 
11  ;  as  9  to  2. 


QUEST. — In  compounding  proportions,  what  may  be  done  with  equal 
factors  or  divisors  ?  When  four  quantities  are  proportional,  if  the  first 
is  greater  than  the  second,  how  is  the  third  ?  If  equal  ?  If  less  ?  If 
four  quantities  are  proportional,  how  are  their  reciprocals? 


Arts.  355-357.]      GEOMETRICAL  PROPORTION. 


219 


Let  x=  the  greater,         and  49 — z=  the  less. 
By  the  conditions  proposed,  z-\-6l3&- — X'.  :9:2 

Adding  terms,  (Art.  350,  2,)  z-|-6:44:  •$' ll 

Dividing  the  consequents,  (Art.  345,  8,)  x+6:4:  :9:1 

Multiplying  the  extremes  and  means,  x+6=36.  And  z— 30. 

Prob.  2.  What  number  is  that,  to  which  if  1,  5,  and  13, 
be  severally  added,  the  first  sum  shall  be  to  the  second,  as 
the  second  to  the  third  ? 

Prob.  3.  Find  two  numbers,  the  greater  of  which  shall  be 
to  the  less,  as  their  sum  to  42 ;  and  as  their  difference  to  6. 

Prob.  4.  Divide  the  number  18  into  two  such  parts,  that 
the  squares  of  those  parts  may  be  in  the  ratio  of  25  to  16. 

Prob.  5.  Divide  the  number  14  into  two  such  parts,  that 
the  quotient  of  the  greater  divided  by  the  less,  shall  be  to  the 
quotient  of  the  less  divided  by  the  greater,  as  16  to  9. 

Prob.  6.  If  the  number  20  be  divided  into  two  parts,  which 
are  to  each  other  in  the  duplicate  ratio  of  3  to  1,  what  num- 
ber is  a  mean  proportional  between  those  parts  ? 

Prob.  7.  There  are  two  numbers  whose  product  is  24,  and 
the  difference  of  their  cubes,  is  to  the  cube  of  their  difference, 
as  19  to  1.  What  are  the  numbers  ? 

Prob.  8.  There  are  two  numbers  in  the  proportion  of  5:6  ; 
the  first  being  increased  by  4  and  the  last  by  6,  the  propor- 
tion will  be  as  4:5.  What  are  the  numbers  ? 

Prob.  9.  A  farmer  has  a  quantity  of  corn  in  his  granary,  and 
sells  a  certain  number  of  bushels,  which  is  to  the  number  of 
bushels  remaining  as  4:5.  He  then  feeds  out  15  bushels, 
which  is  to  the  number  sold  as  1:2.  How  many  bushels  had 
he  at  first,  and  how  many  did  he  sell  ? 

Prob.  10.  There  are  two  numbers  whose  product  is  135, 
and  the  difference  of  their  squares,  is  to  the  square  of  their 
difference,  as  4  to  1.  What  are  the  numbers  ? 


220  ALGEBRA.  [Sect.  XIV. 

Prob.  11.  What  two  numbers  are  those,  whose  difference, 
sum,  and'  product,  are  as  the  numbers  2,  3,  and  5,  respec- 
tively ? 

Prob.  12.  Divide  the  number  24  into  two  such  parts,  that 
their  product  shall  be  to  the  sum  of  their  squares,  as  3  to  10. 

Prob.  13.  In  a  mixture  of  rum  and  brandy,  the  difference 
between  the  quantities  of  each,  is  to  the  quantity  of  brandy, 
as  100  is  to  the  number  of  gallons  of  rum  ;  and  the  same 
difference  is  to  the  quantity  of  rum,  as  4  to  the  number 
of  gallons  of  brandy.  How  many  gallons  are  there  of 
each  ? 

Prob.  14.  There  are  two  numbers  which  are  to  each  other 
as  3  to  2.  If  6  be  added  to  the  greater  and  subtracted  from 
the  less,  the  sum  and  remainder  will  be  to  each  other,  as  3 
to  1.  What  are  the  numbers  ? 

Prob.  15.  There  are  two  numbers  whose  product  is  320  ; 
and  the  difference  of  their  cubes,  is  to  the  cube  of  their  dif- 
ference, as  61  to  1.  What  are  the  numbers  ? 

Prob.  16.  There  are  two  numbers,  which  are  to  each  other, 
in  the  duplicate  ratio  of  4  to  3  ;  and  24  is  a  mean  propor- 
tional between  them.  What  are  the  numbers  ? 

CONTINUED     GEOMETRICAL     PROPORTION 
OR     PROGRESSION. 

358.  When  all  the  ratios  of  a  series  of  proportionals  are 
equal,  the  quantities  are  said  to  be  in  continued  proportion  or 
progression.  (Art.  322.) 

As  arithmetical  proportion  continued  is  arithmetical  progres- 
sion, so  geometrical  proportion  continued  is  geometrical  pro- 
gression. It  is  also  sometimes  called  progression  by  quotient. 

Q.TJEST. — What  is  continued  geometrical  proportion?  What  else  is 
it  called  ? 


Arts.  358-361.]     GEOMETRICAL  PROGRESSION.  221 

The  numbers  64,  32,  16,  8,  4,  are  in  continued  geometri- 
cal proportion.  (Art.  322.)  ^ 

In  this  series,  if  each  preceding  term  be  divided  by  the 
common  ratio,  the  quotient  will  be  the  following  term.  Thus, 

^t— 32,  and  \2-— 16,  and  4£=8,  and  J=4. 
If  the  order  of  the  series  be  inverted,  the   proportion  wilt 
still  be  preserved,  (Art.  344  ;)  and  the  common  divisor  will 
become  a  multiplier.     In  the  series  4,  8,  16,  32,  64,  &c. 
4X2=8,  and  8X2=16,  and  16X2=32,  &c. 

359.  Quantities  then  are  in  geometrical  progression,  when 
they  increase  by  a  common  multiplier,  or  decrease  by  a  com,' 
mon  divisor. 

This  common  multiplier  or  divisor  is  called  the  ratio.  For 
most  purposes,  however,  it  will  be  more  simple  to  consider 
the  ratio  as  always  a  multiplier,  either  integral  or  fractional. 

In  the  series  64,  32,  16,  8,  4,  the  ratio  is  either  2  a  divisor, 
or  J  a  multiplier. 

360.  When  several  quantities  are  in  continued  proportion, 
the  number  of  couplets,  and  of  course  the  number  of  ratios,  is 
one  less  than  the  number  of  quantities.     Thus  the  five  pro- 
portional quantities  a,  b,  c,  d,  e,  form  four  couplets  contain- 
ing four  ratios ;   and  the  ratio  of  a :  e  is  equal  to  the  ratio  of 
a4 :54,  that  is,  the  ratio  of  the  fourth  power  of  the  first  quan- 
tity, to  the  fourth  power  of  the  second.     Hence, 

361.  If  three  quantities  are  proportional,  the  first  is  to  the 
third,  as  the  square  of  the  first  to  the  square  of  the  second  ; 
or  as  the  square  of  the  second,  to  the  square  of  the  third.     In 
other  words,  the  first  has  to  the  third,  a  duplicate  ratio  of  the 
first  to  the  second.     And  conversely,  if  the  first  of  the  three 

QUEST. — When  are  quantities  said  to  be  in  geometrical  progression  ? 
What  is  the  common  multiplier  or  divisor  called  ?     In  a  series  of  con- 
tinued proportion,   how  many  couplets  and   ratios  are  there  ?     When 
there  are  three  proportionals,  what  ratio  has  the  first  to  the  third  ? 
19* 


222  ALGEBRA.  [Sect.  XIV. 

quantities  is  to  the  third,  as  the  square  of  the  first  to  the 
square  of .the  second,  the  three  quantities  are  proportional. 
If  albllblc,  then  alc::a2lb2.     And  universally, 

362.  If  several  quantities  are  in  continued  proportion,  the 
ratio  of  the  first  to  the  last  is  equal  to  one  of  the  intervening 
ratios  raised  to  a  power  whose  index  is  one  less  than  the  num- 
ber of  quantities. 

If  there  are  four  proportionals    a,  J,  c,  d,  then  aid: :  a3 :63 
If  there  are  Jive  a,  £,  c,  d,  e  ;  ale:  la*  :Z>4,  &c. 

363.  If  several  quantities  are  in  continued  proportion,  they 
will  be  proportional  when  the  order  of  the  whole  is  inverted. 
This  has  already  been  proved  with  respect  to  four  propor- 
tional quantities.     (Art.  344,  cor.)     It  may  be  extended  to 
any  number  of  quantities. 

Between  the  numbers,  64,  32,  16,  8,  4, 

The  ratios  are,  2,  2,  2,  2, 

Between  the  same  inverted,  4,  8,  16,  32,  64, 

The  ratios  are,  J,  J,  J,  J. 

So  if  the  order  of  any  proportional  quantities  be  inverted, 
the  ratios  in  one  series  will  be  the  reciprocals  of  those  in  the 
other.  For  by  the  inversion  each  antecedent  becomes  a  con- 
sequent, and  v.  v.,  and  the  ratio  of  a  consequent  to  its  antece- 
dent is  the  reciprocal  of  the  ratio  of  the  antecedent  to  the 
consequent.  (Art.  305.)  That  the  reciprocals  of  equal  quan- 
tities are  themselves  equal,  is  evident  from  Ax.  4. 

364.  To  investigate  the  properties  of  geometrical  progres- 
sion, we  may  take  nearly  the  same  course,  as  in  arithmetical 
progression,  observing  to  substitute  continual  multiplication 
and  division,  instead  of  addition  and  subtraction.     It  is  evi- 
dent, in  the  first  place,  that, 

Q.UEST. — When  several  quantities  are  in  continued  proportion,  what 
ratio  has  the  first  to  the  last  ?  If  the  series  is  inverted,  what  effect  has 
it  ?  How  are  the  ratios  in  one  series,  compared  with  those  of  another, 
when  the  order  is  inverted  ? 


Arts.  362-368.]     GEOMETRICAL  PROGRESSION.  223 

365.  In  an  ascending  geometrical  series,  each  succeeding 
term  is  found,  by  multiplying  the  ratio  into  the  preceding 
term. 

If  the  first  term  is  a,  and  the  ratio  r, 
Then  aXr^ar,  the  second  term,  orXr=or2,  the  third, 

ar2  Xr=ar3,  the  fourth,        or3  Xr=ar4,  the  fifth,  &c. 
And  the  series  is  a,  or,  or2,  ar3,  ar4,  ar5,  &c. 

366.  If  the  first  term  and  the  ratio  are  the  same,  the  pro- 
gression is  simply  a  series  of  powers. 

If  the  first  term  and  the  ratio  are  each  equal  to  r,  , 

Then  rXr=r2,  the  second  term,    r2Xrrr:r3,  the  third, 
r  3  X  r=r  4 ,  the  fourth,  r  4  X  r=r  5 ,  the  fifth. 

And  the  series  is  r,  r2,  r3,  r4,  r5,  r6,  &c. 

367.  In  a  descending  series,  each  succeeding  term  is  found 
by  dividing  the  preceding  term  by  the  ratio,  or  multiplying 
by  the  fractional  ratio. 

If  the  first  term  is  ar6,  and  the  ratio  r, 

6 

The  second  term  is     — ,  or  ar6X^  ; 

And  the  series  is  ar6,  ar5,  ar4,  ar3,  or2,  ar,  a,  &c. 
If  the  first  term  is  a,  and  the  ratio  r, 

The  series  is  a,  — ,  — -,  — ,  &c.  or  a,  ar"1,  ar~2,  &c. 
r   r2    r3 

123  4  5  6 

368.  By  attending  to  the  series  a,  ar,  ar2,  ar3,  ar4,  ar&, 
&c.,  it  will  be  seen  that,  in  each  term,  the  exponent  of  the 
power  of  the  ratio,  is  one  lesst  than  the  number  of  the  term. 

If  then  azr  the  first  term,         r=  the  ratio, 

z—  the  last,  n—  the  number  of  terms  ; 

we  have  the  equation  z— ar71""1,  that  is, 

QUEST. — In  an  ascending  geometrical  series  how  is  each  succeeding 
term  found  ?  When  the  first  term  and  ratios  are  the  same,  what  is  the 
progression  ?  How  is  each  term  found  in  a  descending  series  ? 


224  ALGEBRA.  [Sect.  XIV. 

369.  In  geometrical  progression,  the  last  term  is  equal  to 
the  product  of  the  first,  into  that  power  of  the  ratio  whose 
index  is  one  less  than  the  number  of  terms. 

When  the  least  term  and  the  ratio  are  the  same,  the  equa- 
tion becomes  z=^rrn~1=rn.     (Art.  366.) 

370.  Of  the  four  quantities  a,  z,  r,  and  n,  any  three  being 
given,  the  other  may  be  found. 

1.  By  the  last  article, 

z=rarn~1=the  last  term. 

2.  Dividing  by  r"""1, 

-^Y  =a=  the  first  term. 

3.  Dividing  the  1st  by  a,  and  extracting  the  root, 


/z\«-i 

u    =r= 


the  ratio. 


371.  By  the  last  equation  may  be  found  any  number  of 
geometrical  means,  between  two  given  numbers.     If  m=  the 
number  of  means,  m-|-2— n,  the  whole  number  of  terms. 
Substituting  m+2,  for  n,  in  the  equation,  we  have 

i 

fz\mJ*-i 

=zr,  the  ratio. 

\a] 

When  the  ratio  is  found,  the  means  are  obtained  by  con- 
tinued multiplication. 

Prob.  1.  Find  two  geometrical  means  between  4  and  256. 
Ans.  The  ratio  is  4,  and  the  series  is  4,  16,  64,  256. 

Prob.  2.  Find  three  geometrical  means  between  £  and  9. 

372.  The  next  thing  to  be  attended  to,  is  the  rule  for  find- 
ing the  sum  of  all  the  terms. 

QUEST. — What  is  the  last  term  equal  to  ?  What  is  the  first  term 
equal  to  ?  How  find  the  ratio  ? 


Arts.  369-372.]     GEOMETRICAL  PROGRESSION.  225 

If  any  term,  in  a  geometrical  series,  be  multiplied  by  the 
ratio,  the  product  will  be  the  succeeding  term.  (Art.  365.) 
Of  course,  if  each  of  the  terms  be  multiplied  by  the  ratio,  a 
new  series  will  be  produced,  in  which  all  the  terms  except 
the  last  will  be  the  same,  as  all  except  the  first  in  the  other 
series.  To  make  this  plain,  let  the  new  series  be  written 
under  the  other,  in  such  a  manner,  that  each  term  shall  be 
removed  one  step  to  the  right  of  that  from  which  it  is  pro- 
duced in  the  line  above. 

Take,  for  instance,  the  series  2,  4,  8,  16,  32 

Multiplying  each  term  by  the  ratio,  4,  8,  16,  32,  64 

Here  it  will  be  seen  at  once,  that  the  four  last  terms  in  the 
upper  line  are  the  same,  as  the  four  first  in  the  lower  line. 
The  only  terms  which  are  not  in  loth^  are  the  first  of  the  one 
series,  and  the  last  of  the  other.  So  that  when  we  subtract 
the  one  series  from  the  other,  all  the  terms  except  these  two 
will  disappear,  by  balancing  each  other. 

If  the  given  series  is            a,  ar,  ar2,  ar3,  ....  ar""""1. 
Then  mult,  by  r,  we  have       ar,  ar2,  ar3, ar*"1,  arn. 

Now  let  s=  the  sum  of  the  terms. 

Then  snra+ar+ar^-ar3, +arn~1, 

And  mult,  by  r,    rs=     ar+ar2+ar3,  ....  +ar*—1+arn. 

Subt.  the  first  equation  from  the  second,       rs — s=arn — a 

arn — a 


And  dividing  by  (r— 1,)  (Art.  98,)         s= 


r— 1 


In  this  equation,  arn  is  the  last  term  in  the  new  series,  and 
is  therefore  the  product  of  the  ratio  into  the  last  term  in  the 
given  series. 

Therefore,    s= — ,  that  is, 

r — 1 


ALGEBRA.  [Sect.  XIV. 

373.  To  find  the  sum  of  a  geometrical  series, 

Multiply  the  last  term  into  the  ratio,  from  it  subtract  the 
first  term,  and  divide  the  remainder  by  the  ratio  less  one. 

Obser.  From  the  above  formula,  in  connexion  with  the  one  in  Art. 
368,  there  may  be  the  same  variety  of  other  formulas  deduced  as  in 
Art.  333.  The  others  however  involve  principles  with  which,  it  is 
presumed,  the  pupil  is  not  yet  acquainted. 

Prob.  3.  If  in  a  series  of  numbers  in  geometrical  progres- 
sion, the  first  term  is  6,  the  last  term  1458,  and  the  ratio  3, 
what  is  the  sum  of  all  the  terms  ? 

rz— a      3X1458—6 

Ans.  s— -  =  -      — - —2184. 

r — 1  o — 1 

Prob.  4.  If  the  first  term  of  a  decreasing  geometrical  se- 
ries is  J,  the  ratio  J,  and  the  number  of  terms  5 ;  what  is  the 
sum  of  the  series  ? 

Prob.  5.  What  is  the  sum  of  the  series,  1,  3,  9,  27,  &c.  to 
12  terms  ? 

Prob.  6.  What  is  the  sum  of  ten  terms  of  the  series  1,  f , 
|,  jfr,  &c. 

Prob.  7.  If  the  first  term  of  a  series  is  2,  the  ratio  2,  and 
the  number  of  terms  13  ;  ^hat  is  the  last  term  ? 

Prob.  8.  What  is  the  12th  term  of  a  series,  the  1st  term  of 
which  is  3,  and  the  ratio  3  ? 

Prob.  9.  A  man  bought  a  horse,  giving  1  cent  for  the  first 
nail  in  his  shoes,  three  for  the  second,  and  so  on.  The  shoes 
contained  32  nails  ;  what  was  the  cost  of  the  horse  ? 

374.  Quantities   in  geometrical  progression  are  propor- 
tional to  their  differences. 

QUEST. — How  is  the  sum  of  a  geometrical  series  found?  If  quan- 
tities are  in  geometrical  progression,  what  is  true  of  their  differ- 
ence ? 


Arts.  373-375.]     GEOMETRICAL  PROGRESSION.  227 

Let  the  series  be  a,  ar,  ar2,  ar3,  ar*,  &c- 
By  the  nature  of  geometrical  progression, 

alar',  \ar\ar2'.  Iar2lar3l  :ar3:ar4,  &c. 
In  each  couplet  let  the  antecedent  be  subtracted  from  the 
consequent,  according  to  Art.  350,  6. 

Then  alarl  lar — alar2 — arl  lar2 — ar:ar3 — ar2,  &c. 

That  is,  the  first  term  is  to  the  second,  as  the  difference 
between  the  first  and  second,  to  the  difference  between  the 
second  and  third ;  and  as  the  difference  between  the  se- 
cond and  third,  to  the  difference  between  the  third  and 
fourth,  &c. 

Cor.  If  quantities  are  in  geometrical  progression,  their  dif> 
ferences  are  also  in  geometrical  progression. 

Thus  the  numbers  3,  9,    27,     81,      243,  &c. 

And  their  differences         6,  18,     54,     162,   &c.      are  in 

geometrical  progression. 

375.  Problems  in  geometrical  progression,  may  be  solved, 
as  in  other  parts  of  algebra,  by  means  of  equations. 

Prob.  10.  Find  three  numbers  in  geometrical  progression, 
such  that  their  sum  shall  be  14,  and  the  sum  of  their 
squares  84. 

Let  the  three  numbers  be  z,  y,  and  z. 
By  the  conditions,  xly  I  lylz,  or  xz=y2 

And  x-\-y-{-z=  14 

And  X2_|^2^2_.84 

Ans.  2,  4  and  8. 

Prob.  11.  There  are  three  numbers  in  geometrical  progres- 
sion whose  product  is  64,  and  the  sum  of  their  cubes  is  584. 
What  are  the  numbers  ? 

Prob.  12.  There  are  three  numbers  in  geometrical  progres- 
sion :  the  sum  of  the  first  and  last  is  52,  and  the  square  of 
the  mean  is  100.  What  are  the  numbers  ? 


228  ALGEBRA.  [Sect.  XV. 

Prob.  13.  Of  four  numbers  in  geometrical  progression,  the 
sum  of  the  two  first  is  15,  and  the  sum  of  the  two  last  is  60. 
What  are  the  numbers  ? 

Prob.  14.  A  gentleman  divided  210  dollars  among  three 
servants,  in  such  a  manner,  that  their  portions  were  in  geo- 
metrical progression  ;  and  the  first  had  90  dollars  more  than 
the  last.  How  much  had  each  ? 

Prob.  15.  There  are  three  numbers  in  geometrical  progres- 
sion, the  greatest  of  which  exceeds  the  least  by  15  ;  and  the 
difference  of  the  squares  of  the  greatest  and  the  least,  is  to 
the  sum  of  the  squares  of  all  the  three  numbers  as  5  to  7. 
What  are  the  numbers  ? 

Prob.  16.  There  are  four  numbers  in  geometrical  progres- 
sion, the  second  of  which  is  less  than  the  fourth  by  24  ;  and 
the  sum  of  the  extremes  is  to  the  sum  of  the  means,  as  7  to  3. 
What  are  the  numbers  ? 


SECTION    XV. 

EVOLUTION  OF  COMPOUND  QUANTITIES. 

376.  RULE. — Arrange  the  terms  according  to  the  powers 
of  one  of  the  letters,  so  that  the  highest  power  shall  stand 
first,  the  next  highest  next,  &c. 

Take  the  root  of  the  first  term,  for  the  first  term  of  the 
required  root : 

Subtract  the  power  from  the  given  quantity,  and  divide  the 
first  term  of  the  remainder,  by  the  first  term  of  the  root  in- 

QUEST. — How  should  the  terms  be  arranged  to  extract  the  root  of  a 
compound  quantity  ?  What  are  the  other  steps  ? 


Art.  376.]     EVOLUTION  OF  COMPOUND  QUANTITIES.          229 

volved  to  the  next  inferior  power,  and  multiplied  by  the  index 
of  the  given  power  ;*  the  quotient  will  be  the  next  term  of  the 
root. 

Subtract  the  power  of  the  terms  already  found  from  the 
given  quantity,  and  using  the  same  divisor,  proceed  as  before. 

This  rule  verifies  itself.  For  the  root,  whenever  a  new 
term  is  added  to  it,  is  involved,  for  the  purpose  of  subtracting 
its  power  from  the  given  quantity  :  and  when  the  power  is 
equal  to  this  quantity,  it  is  evident  the  true  root  is  found. 

1.  Extract  the  cube  root  of 


a6,  the  first  subtrahend. 


3«4)*    3ab,  &c.  the  first  remainder. 

a6+3a5+3a4+a3,  the  2d  subtrahend. 
3a4)  *    *     —  6a4,  &c.  the  2d  remainder. 


2.  Find  the  4th  root  of  a4+8a3+24a2+32a+16. 

3.  Find  the  5th  root  of  a5  +  5«46  +  10a362  +  10a263 


4.  Find  the  cube  root  of  a3  —  6a2b+2ab2  —  863. 

5.  Find  the  2d  root  of  4a2  —  12ab+9b2  +  IGah—  24bh 
+  16A2. 

N.  B.  In  finding  the  divisor  in  the  5th  example,  the  term 
2«  in  the  root  is  not  involved,  because  the  power  next  below 
the  square  is  the  first  power. 

*  By  the  given  power  is  meant  a  power  of  the  same  name  with  the 
required  root.  As  powers  and  roots  are  correlative  terms,  any  quantity 
is  the  square  of  its  square  root,  the  cube  of  its  cube  root,  &c. 

20 


230  ALGEBRA.  [Sect.  XV. 

377.  The  square  root  may  be  extracted  by  the  following 

RULE. — Arrange  the  terms  according  to  the  powers  of  one 
of  the  letters,  take  the  root  of  the  first  term,  for  the  first  term 
of  the  required  root,  and  subtract  the  power  from  the  given 
quantity. 

Bring  down  two  other  terms  for  a  dividend.  Divide  by 
double  the  root  already  found,  and  add  the  quotient  both  to 
the  root,  and  to  the  divisor.  Multiply  the  divisor  thus  in- 
creased, into  the  term  last  placed  in  the  root,  and  subtract  the 
product  from  the  dividend. 

Bring  down  two  or  three  additional  terms  and  proceed  as 
before. 

PROOF. — Multiply  the  root  into  itself,  and  if  the  product 
is  equal  to  the  given  quantity,  the  work  is  right. 

6.  What  is  the  square  root  of 

a2+2ab+b2+2ac+2bc+c2(a+b+c 
a2,  the  first  subtrahend. 
2a+b)  *     2ab+b2 
Into  6=r       2ab-\-b2,  the  second  subtrahend. 

2a+2b+c)    *      *    2ac+2bc+c2 

Into  c=  2ac-{-2bc-\-c2 ,  the  third  subtrahend. 

Proof.  The  square  of  the  root  a-\-b-\-c,  is  equal  to  the 
given  quantity. 

For  (a+b)2=ia2+2ab+b2=:a2+(2a+b)Xb.    (Art.  97.) 

And  substituting  h=a+b,  the  square  h2=a2+(2a+b)Xb. 

And  (a+b+c)2=(h+c)2=h2+(2h+c)Xc ; 
that  is,  restoring  the  values  of  h  and  h2, 

(a+b+c)2=a2+(2a+b)Xb+(2a+2b+c)Xc. 

In  the  same  manner,  it  may  be  proved,  that,  if  another 
term  be  added  to  the  root,  the  power  will  be  increased,  by 
the  product  of  that  term  into  itself,  and  into  twice  the  sum  of 
the  preceding  terms. 

QUEST. — What  is  the  rule  for  extracting  the  square  root? 


Arts.  377,  378.]   EVOLUTION  OF  COMPOUND  QUANTITIES.     231 

The  demonstration  will  be  substantially  the  same,  if  some 
of  the  terms  be  negative. 

7.  Find  the  square  root  of  1  —  46+462+%  —  ^ty+y2- 

8.  Find  the  square  root  of  a6  —  2a5+3a4  —  2a3+a2. 

9.  Find  the  square  root  of  «4+4a26+462—  4a2  —  86+4. 
378.  It  will   frequently  facilitate  the  extraction  of  roots,  to 

consider  the  index  as  composed  of  two  or  more  factors. 

Thus  «i~a*X*.     And  a^=a^X^.     That  is, 

The  fourth  root  is  equal  to  the  square  root  of  the  square 
root  ; 

The  sixth  root  is  equal  to  the  square  root  of  the  cube  root  ; 

The  eighth  root  is  equal  to  the  square  root  of  the  fourth 
root,  &c. 

To  find  the  sixth  root,  therefore,  we  may  first  extract  the 
cube  root,  and  then  the  square  root  of  this. 

10.  Find  the  square  root  of  x4  —  4x3-|-6x2  —  4x-)-l. 

11.  Find  the  cube  root  of  x6  —  6x5+15x*  —  20x3+15z2 
—  6x+l. 

12.  Find  the  square  root  of  4x4—  4x3+13x2—  6x+9. 

13.  Find  the  4th  root  of  16a4—  96a3x+216«2x2—  216ax3 


14.  Find  the  5th  root  of  x5+5x4+10x3+10x2+5x+l. 

15.  Find  the  6th  root  of  a6  —  6a56  +  15a±62  —  20a3&3 


QUEST.  —  How  may  the  extraction  of  roots  be  facilitated  ?  What  is 
the  fourth  root  equal  to  ?  The  sixth  ?  The  eighth  ?  How  then  ma/ 
we  find  the  sixth  root  ? 


232  ALGEBRA.'  [Sect.  XVI. 

SECTION    XVI. 


ART.  379.  It  is  often  expedient  to  make  use  of  the  alge- 
braic notation,  for  expressing  the  relations  of  geometrical 
quantities,  and  to  throw  the  several  steps  of  a  demonstration 
into  the  form  of  equations.  By  this,  the  nature  of  the  reason- 
ing is  not  altered.  It  is  only  translated  into  a  different  lan- 
guage. Signs  are  substituted  for  words,  but  they  are  intend- 
ed to  convey  the  same  meaning.  A  great  part  of  the  de- 
monstrations in  geometry,  really  consist  of  a  series  of  equa- 
tions, though  they  may  not  be  presented  to  us  under  the  alge- 
braic forms.  Thus  the  proposition,  that  the  sum  of  the  three 
angles  of  a  triangle  is  equal  to  two 
right  angles,  (Euc.  32,  1,)  may  be 
demonstrated,  either  in  common  lan- 
guage, or  by  means  of  the  signs  used 
in  algebra. 

Let  the  side  AB,  of  the  triangle 
ABC,  (Fig.  1,)  be  continued  to  D ; 
let  the  line  BE  be  parallel  to  AC ;   A 
and  let  GHI  be  a  right  angle. 

The  demonstration,  in  words,  is  as  follows : 

1.  The  angle  EBD  is  equal  to  the  angle  BAG,  (Euc.  29,  1.) 

2.  The  angle  CBE  is  equal  to  the  angle  ACB. 

3.  Therefore,  the  angle   EBD  added  to  CBE,  that  is,  the 

angle  CBD,  is  equal  to  BAG  added  to  ACB. 

4.  If  to  these  equals,  we  add  the  angle  ABC,  the  angle  CBD 

added  to  ABC,  is  equal  to  BAG  added  to  ACB  and  ABC. 

*  This  section  is  to  be  read  after  the  Elements  of  Geometry. 


Arts.  379-381.]    APPLICATION  TO  GEOMETRY.  233 

5.  But  CBD  added  to  ABC,  is  equal  to  twice  GHI,  that  is,  to 

two  right  angles.     (Euc.  13.  1.) 

6.  Therefore,  the  angles  BAG,  and  ACB,  and  ABC,  are  to- 

gether equal  to  twice  GHI,  or  two  right  angles. 
Now  by  substituting  the  sign  -f-,  for  the  word  added,  or  and, 
and  the  sign  =,  for  the  word  equal,  we  shall  have  the  same 
demonstration  in  the  following  form. 

1.  By  Euclid  29.  1.  EBDz=BAC 

2.  And  CBE^ACB 

3.  Add  the  two  equations  EBD+CBEz=BAC+ACB 

4.  Add  ABC  to  both  sides  CBD-f  ABC=BAC+ACB+ABC 

5.  But  by  Euclid  13.  1.     CBD+ABC=:2GHI 

6.  Make  the  4th&  5th  equal  B  AC+ACB+ ABC=2GHI. 

By  comparing,  one  by  one,  the  steps  of  these  two  demon- 
strations, it  will  be  seen,  that  they  are  precisely  the  same,  ex- 
cept that  they  are  differently  expressed. 

380.  It  will  be  observed  that  the  notation  in  the  example 
just  given,  differs,  in  one  respect,  from  that  which  is  general- 
ly used  in  algebra.     Each  quantity  is  represented,  not  by  a 
single  letter,  but  by  several.     In  common  algebra  when  one 
letter  stands  immediately  before  another  as  ab,  without  any 
character  between  them,  they  are  to  be  considered  as  multi- 
plied together. 

But  in  geometry,  AB  is  an  expression  for  a  single  line,  and 
not  for  the  product  of  A  into  B.  Multiplication  is  denoted, 
either  by  a  point  or  by  the  sign  X  •  The  product  of  AB  into 
CD,  is  AB.CD,  or  ABxCD. 

381.  There  is  no  impropriety,  however,  in  representing  a 
geometrical  quantity  by  a  single   letter.     We   may  make  b 
stand  for  a  line  or  an  angle,  as  well  as  for  a  number. 

20* 


234  ALGEBRA.  [Sect.  XVI. 

If,  in  the  example  above,  we  put  the  angle 

«,  ACB=</,  ABC=A, 

=  b,  CBDr=^,  GHI  =1, 

CBErzzc, 

the  demonstration  will  stand  thus ; 

1.  By  Euclid,  29,  1,  a=6 

2.  And  c—d 

3.  Adding  the  two  equations,       a-\-c=g=b-\-d 

4.  Adding  h  to  both  sides,  g-{-h=b-\-d-{-h 

5.  By  Euclid,  13,  1, 

6.  Making  the  4th  and  5th  equal, 

This  notation  is  apparently  more  simple  than  the  other ; 
but  it  deprives  us  of  what  is  of  great  importance  in  geometri- 
cal demonstrations,  a  continual  and  easy  reference  to  the  fig- 
ure. To  distinguish  the  two  methods,  capitals  are  generally 
used,  for  that  which  is  peculiar  to  geometry ;  and  small  letters, 
for  that  which  is  properly  algebraic. 

382.  If  a  line  whose  length  is  meas-  Fig.  2. 
ured  from  a  given  point  or  line,  be 
considered  positive;  a  line  proceed- 
ing in  the  opposite  direction  is  to  be 
considered  negative.     If  AB  (Fig.  2,) 

reckoned  from  DE  on  the  right,  is 
positive;  AC  on  the  left  is  nega- 
tive. 

Hence,  if  in  the  course  of  a  calculation,  the  algebraic  value 
of  a  line  is  found  to  be  negative ;  it  must  be  measured  in  a 
direction  opposite  to  that  which,  in  the  same  process,  has  been 
considered  positive.  (Art.  162. a.) 

383.  In  algebraic  calculations,  there  is  frequent  occasion 
for  multiplication,  division,  involution,  &c.     But  how,  it  may 


B 


Arts.  382-386.]  APPLICATION  TO  GEOMETRY.  235 

be  asked,  can  geometrical  quantities  Jje  multiplied  into  each 
other  ?  One  of  the  factors,  in  multiplication,  is  always  to  be 
considered  as  a  number.  The  operation  consists  in  repeating 
the  multiplicand  as  many  times  as  there  are  units  in  the  mul- 
tiplier. How  then  can  a  line,  a  surface,  or  a  solid,  become 
a  multiplier  ? 

To  explain  this,  it  will  be  necessary  to  observe,  that  when- 
ever one  geometrical  quantity  is  multiplied  into  another,  some 
particular  extent  is  to  be  considered  the  unit.  It  is  immate- 
rial what  this  extent  is,  provided  it  remains  the  same,  in  dif- 
ferent parts  of  the  same  calculation.  It  may  be  an  inch,  a 
foot,  a  rod,  or  a  mile.  If,  for  instance,  one  of  the  lines  be  a 
foot  long,  and  the  other  half  a  foot ;  the  factors  will  be,  one 
12  inches,  and  the  other  6,  and  the  product  will  be  72  inches. 
Though  it  would  be  absurd  to  say  that  one  line  is  to  be  re- 
peated as  often  as  another  is  long ;  yet  there  is  no  impro- 
priety in  saying,  that  one  is  to  be  repeated  as  many  times,  as 
there  are  feet  or  rods  in  the  other.  This,  the  nature  of  a 
calculation  often  requires. 

384.  If  the  line  which  is  to  be  the  multiplier,  is  only  a  part 
of  the  length  taken  for  the  unit ;  the  product  is  a  like  part  of 
the  multiplicand.     (Art.  71.)     Thus,  if  one  of  the  factors  is 
6  inches,  and  the  other  half  an  inch,  the  product  is  3  inches. 

385.  Instead  of  referring  to  the  measures  in  common  use, 
as  inches,  feet,  &c.  it  is  often  convenient  to  fix  upon  one  of 
the  lines  in  a  figure,  as  the  unit  with  which  to  compare  all  the 
others.     When  there  are  a  number  of  lines  drawn  within  and 
about  a  circle,  the  radius  is  commonly  taken  for  the   unit. 
This  is  particularly  the  case  in  trigonometrical  calculations. 

386.  The  observations  which  have  been  made  concerning 
lines,  may  be  applied  to  surfaces  and  solids.     There  may  be 
occasion  to  multiply  the  area  of  a  figure,  by  the  number  of 
inches  in  some  given  line. 


236  ALGEBRA.  [Sect.  XVI, 

But  here  another  difjculty  presents  itself.  The  product  of 
two  lines  is  often  spoken  of  as  being  equal  to  a  surface ;  and 
the  product  of  a  line  and  a  surface,  as  equal  to  a  solid.  But 
if  a  line  has  no  breadth,  how  can  the  multiplication,  that  is, 
the  repetition,  of  a  line  produce  a  surface  ?  And  if  a  sur- 
face has  no  thickness,  how  can  a  repetition  of  it  produce  a 
solid  ? 

387.  In  answering  these  inquiries,  it  must  be  admitted,  that 
measures  of  length  do  not  belong  to  the  same  class  of  mag- 
nitudes with  superficial  or  solid  measures ;  and  that  none  of 
the  steps  of  a  calculation  can,  properly  speaking,  transform 
the  one  into  the  other.  But,  though  a  line  cannot  become  a 
surface  or  a  solid,  yet  the  several  measuring  units  in  common 
use  are  so  adapted  to  each  other,  that  squares,  cubes,  &c.  are 
bounded  by  lines  of  the  same  name.  Thus  the  side  of  a 
square  inch,  is  a  linear  inch  ;  that  of  a  square  rod,  a  linear 
rod,  &c.  The  length  of  a  linear  inch  is,  therefore,  the  same 
as  the  length  or  breadth  of  a  square  inch. 

If  then  several  square  inches  are  Fig-  3. 

placed  together,  as  from  Q  to  R, 
(Fig.  3,)  the  number  of  them  in  the 
parallelogram  OR  is  the  same  as  the 
number  of  linear  inches  in  the  side 
QR :  and  if  we  know  the  length  of 
this,  we  have  of  course  the  area  of 
the  parallelogram,  which  is  here  ITT 
supposed  to  be  one  inch  wide. 

But,  if  the  breadth  is  several  inches,  the  larger  parallelo- 
gram contains  as  many  smaller  ones,  each  an  inch  wide,  as 
there  are  inches  in  the  whole  breadth.  Thus,  if  the  paral- 
lelogram AC,  (Fig.  3,)  is  5  inches  long,  and  3  inches  broad, 
it  may  be  divided  into  three  such  parallelograms  as  OR.  To 
obtain,  then,  the  number  of  squares  in  the  large  parallelo- 


Arts.  387-389.]  APPLICATION  TO  GEOMETRY.  237 

gram,  we  have  only  to  multiply  the  number  of  squares  in  one 
of  the  small  parallelograms,  into  the  number  of  such  paral- 
lelograms contained  in  the  whole  figure.  But  the  number  of 
square  inches  in  one  of  the  small  parallelograms  is  equal  to 
the  number  of  linear  inches  in  the  length  AB.  And  the  num- 
ber of  small  parallelograms,  is  equal  to  the  number  of  linear 
inches  in  the  breadth  BC.  It  is  therefore  said  concisely,  that 
the  area  of  the  parallelogram  is  equal  to  the  length  multiplied 
into  the  breadth. 

388.  We  hence  obtain  a  convenient  algebraic  expression, 
for  the  area  of  a  right-angled  parallelogram.  If  two  of  the 
sides  perpendicular  to  each  other  are  AB  and  BC,  the  expres- 
sion for  the  area  is  AB  X  BC ;  that  is,  putting  a  for  the  area, 


It  must  be  remarked,  however,  that  when  AB  stands  for 
a  line,  it  contains  only  linear  measuring  units ;  but  when  it 
enters  into  the  expression  for  the  area,  it  is  supposed  to  con- 
tain superficial  units  of  the  same  name. 

389.  The  expression  for  the  area  Fig-  4. 

may  also  be  derived  by  another  me- 
thod more  simple,  but  less  satisfac- 
tory perhaps  to  some.  Let  a  (Fig. 
4,)  represent  a  square  inch,  foot, 
rod,  or  other  measuring  unit ;  and 
let  b  and  I  be  two  of  its  sides.  Also, 
let  A  be  the  area  of  any  right-angled 
parallelogram,  B  its  breadth,  and  L  its  length.  Then  it  is 
evident,  that,  if  the  breadth  of  each  were  the  same,  the  areas 
would  be  as  the  lengths ;  and,  if  the  length  of  each  were  the 
same,  the  areas  would  be  as  the  breadths. 

That  is,         A:a:  :L:?,  when  the  breadth  is  given  ; 
And  A: a:  :B:5,  when  the  length  is  given  ; 

Therefore,  A: a:  :BxL:W,  when  both  vary. 
That  is,  the  area  is  as  the  product  of  the  length  and  breadth. 


238 


ALGEBRA. 


[Sect.  XVI. 


390.  Hence,  in  quoting  the  elements  of  geometry,  the  term 
product  is  frequently  substituted  for  rectangle.     And  what- 
ever is  there  proved  concerning  the  equality  of  certain  rect- 
angles, may  be  applied  to  the  product  of  the  lines  which  con- 
tain the  rectangles.     (Legendre,  166.) 

391.  The  area  of  an  oblique  parallelogram  is  also  obtained, 
by  multiplying  the  base  into  the  perpendicular  height.     Thus 
the  expression  for  the  area  of  the  parallelogram  ABNM  (Fig. 
5,)  is  MNXAD  or  ABxBC.     For  by  Art.  388,  ABxBC  is 
the  area  of  the  right-angled  parallelogram  ABCD ;  and  by 
Euclid  36,  1,  parallelograms  upon  equal  bases,  and  between 
the  same   parallels,  are  equal ;  that  is,  ABCD  is  equal  to 
ABNM. 

Fig.  5.  Fig.  6. 

D  C 

M       D  N         C 


B 


392.  The  area  of  a  square  is  obtained,  by  multiplying  one 
of  the  sides  into  itself.     Thus  the  expression  for  the  area  of 
the  square  AC,  (Fig.  6,)  is  (AB)2,  that  is,  a=(AB)2. 

For  the  area  is  equal  to  ABxBC.     (Art.  388.) 

But  AB=BC,  therefore,  ABxBC=ABxAB=z(AB)2. 

393.  The  area  of  a  triangle  is  equal  to  half  the  product  of 
the  base  and  height.     Thus  the  area  of  the  triangle  ABG, 
(Fig.  7,)  is  equal  to  half  AB  into  GH  or  its  equal  BC,  that  is, 


For  the  area  of  the   parallelogram  ABCD  is  ABxBC, 
(Art.  388.)     And  by  Euclid  41,  1,  if  a  parallelogram  and  a 


Arts.  390-395.]  APPLICATION  TO  GEOMETRY. 


239 


triangle  are  upon  the  same  base,  and  between  the  same  par- 
allels, the  triangle  is  half  the  parallelogram.    (Leg.  168.) 

394.  Hence,  an  algebraic  expression  may  be  obtained  for 
the  area  of  any  figure  whatever,  which  is  bounded  by  right 
lines.  For  every  such  figure  may  be  divided  into  triangles. 

Fig.  7.  Fig.  8.  D 


Thus  the  right-lined  figure,  ABODE  (Fig.  8,)  is  composed 
of  the  triangles  ABC,  ACE,  and  BCD. 

The  area  of  the  triangle  ABCrrJACxBL, 

That  of  the  triangle  ACE=JACxEH, 

That  of  the  triangle  ECD^rJECxDG. 

The  area  of  the  whole  figure  is,  therefore,  equal  to 
(JACXBL)+(JACXEH)+(JECXDG). 
395.  The  expression  for  the  superficies  has  here  been  de- 
rived from  that  of  a  line  or  lines.     It  is  frequently  necessary 
to  reverse  this  order ;  to  find  a  side  of  a  figure,  from  knowing 
its  area. 

If  the  number  of  square  inches  in  the  parallelogram  ABCD 
(Fig.  3,)  whose  breadth  BC  is  3  inches,  be  divided  by  3 ; 
the  quotient  will  be  a  parallelogram,  ABEF,  one  inch  wide, 
and  of  the  same  length  with  the  larger  one.  But  the  length 
of  the  small  parallelogram,  is  the  length  of  its  side  AB.  The 
number  of  square  inches  in  one  is  the  same,  as  the  number 
of  linear  inches  in  the  other.  (Art.  387.)  If  therefore,  the 
area  of  the  large  parallelogram  be  represented  by  a,  the 


240  ALGEBRA.  [Sect.  XVI. 

side  AB:=-=^-,  that  is,  the    length  of  a  parallelogram   is 
BL> 

found  by  dividing  the  area  by  the  breadth. 

If  a  be  put  for  the  area  of  a  square  whose  side  is  AB, 

Then  by  Art.  392,  a=(AB)2, 

And  extracting  both  sides  \/a=AB. 

That  is,  the  side  of  the  square  is  found,  by  extracting  the 

square  root  of  the  number  of  measuring  units  in  its  area. 
396.  If  AB  be  the  base  of  a  triangle  and  BC  its  perpen- 

dicular height  ; 

Then  by  Art.  393,  « 


And  dividing  by  JBC,  -       —  AB. 


That  is,  the  base  of  a  triangle  is  found,  by  dividing  the 
area  by  half  the  height. 

397.  As  a  surface  is  expressed,  by  the  product  of  its  length 
and  breadth  ;    the  contents  of  a  solid  may  be  expressed,  by 
the  product  of  its  length,  breadth  and  depth.     It  is  necessary 
to  bear  in  mind,  that  the  measuring  unit  of  solids,  is  a  cube  ; 
and  that  the  side  of  a  cubic  inch,  is  a  square  inch  ;   the  side 
of  a  cubic  foot,  a  square  foot,  &c. 

Let  ABCD  (Fig.  3,)  represent  the  base  of  a  parallelepiped, 
five  inches  long,  three  inches  broad,  and  one  inch  deep.  It 
is  evident  there  must  be  as  many  cubic  inches  in  the  solid, 
as  there  are  square  inches  in  its  base.  And,  as  the  product 
of  the  lines  AB  and  BC  gives  the  area  of  this  base,  it  gives, 
of  course,  the  contents  of  the  solid.  But  suppose  that  the 
depth  of  the  parallelepiped,  instead  of  being  one  inch,  is  four 
inches.  Its  contents  must  be  four  times  as  great.  If,  then, 
the  length  be  AB,  the  breadth  BC,  and  the  depth  CO,  the 
expression  for  the  solid  contents  will  be,  ABxBCxCO. 

398.  By  means  of  the   algebraic  notation,  a  geometrical 
demonstration  may  often  be  rendered  much  more  simple  and 


.  396-399.]     APPLICATION  TO  GEOMETRY.  241 

concise,  than  in  ordinary  language.  The  proposition,  (Euc. 
4.  2.)  that  when  a  straight  line  is  divided  into  two  parts,  the 
square  of  the  whole  line  is  equal  to  the  squares  of  the  two 
parts,  together  with  twice  the  product  of  the  parts,  is  demon- 
strated, by  involving  a  binomial. 

Let  the  side  of  a  square  be  represented  by  s  ; 

And  let  it  be  divided  into  two  parts,  a  and  b. 

By  the  supposition,  s=a-{-b 

And  squaring  both  sides,  s2=a2-}-2ab-\-b2. 

That  is,  s2  the  square  of  the  whole  line,  is  equal  to  a2  and 
62,  the  squares  of  the  two  parts,  together  with  2a6,  twice  the 
product  of  the  parts. 

399.  The  algebraic  notation  may  also  be  applied,  with 
great  advantage,  to  the  solution  of  geometrical  problems.  In 
doing  this,  it  will  be  necessary,  in  the  first  place,  to  raise  an 
algebraic  equation  from  the  geometrical  relations  of  the 
quantities  given  and  required  ;  and  then  by  the  usual  reduc- 
tions, to  find  the  value  of  the  un-  Fig.  9. 
known  quantity  in  this  equation. 

Prob.  1.  Given  the  base,  and  the 
sum  of  the  hypothenuse  and  per- 
pendicular, of  the  right  angled  tri- 
angle ABC,  (Fig.  9,)  to  find  the 
perpendicular. 

Let  the  base  AB=b 

The  perpendicular  I$C=x 

The  sum  of  hyp,  and  perp.       x-\-AC=a 
Then  transposing  z,  AC=a — x 

1.  By  Euclid  47.  1,  (BC)2+(AB)2:=(AC)2 

2.  That  is,  by  the  notation,  x2-f62—(a-x)2=«2-2ax+x2. 

a2 £2 

And,     x=  — =BC,  the  side  required.     Hence, 

21 


242 


ALGEBRA. 


[Sect.  XVI. 


4  In  a  right  angled  triangle,  the  perpendicular  is  equal  to 
the  square  of  the  sum  of  the  hypothenuse  and  perpendicular, 
diminished  by  the  square  of  the  base,  and  divided  by  twice 
the  sum  of  the  hypothenuse  and  perpendicular.' 

'  It  is  applied  to  particular  cases  by  substituting  numbers  for 
the  letters  a  and  b.  Thus  if  the  base  is  8  feet,  and  the  sum 
of  the  hypothenuse  and  perpendicular  16,  the  expression 

2        7*2  1  f\2        Q2 

— ,  becomes  — — —?-  ~6,  the  perpendicular  ;   and  this 

lid  .  <^X  lu 

subtracted  from  16,  the  sum  of  the  hypothenuse  and  perpen- 
dicular, leaves  10,  the  length  of  the  hypothenuse. 

Prob.  2.  Given  the  base  and  the  difference  of  the  hypothe- 
nuse and  perpendicular,  of  a  right  angled  triangle,  to  find  the 
perpendicular. 

Fig.  10.  Fig.  11. 

C  DC 


Let  the  base,  (Fig.  10,) 
The  perpendicular 
The  given  difference, 
Then  will  the  hypothenuse 

1.  Then  by  Euclid  47.  1, 

2.  That  is,  by  the  notation, 

3.  Expanding  (x+d)2, 

4.  Therefore 


AB—  6rr 


AC:rr; 

(AC)2=(AB)2+(BC)2 


b2—d2 


Prob.  3.  If  the  hypothenuse  of  a  right  angled  triangle  is 
30  feet,  and  the  difference  of  the  other  two  sides  6  feet,  what 
is  the  length  of  the  base  ? 


Art.  399.)  APPLICATION  TO  GEOMETRY. 


243 


Prob.  4.  If  the  hypothenuse  of  a  right  angled  triangle  is 
50  rods,  and  the  base  is  to  the  perpendicular  as  4  to  3,  what 
is  the  length  of  the  perpendicular  ? 

Prob.  5.  Having  the  perimeter  and  the  diagonal  of  a  par- 
allelogram ABCD,  (Fig.  11,)  to  find  the  sides. 

Let  the  diagonal  AC:r=7i— 10 

The  side  AB=z 

Half  the  perimeter  BC+AB=BC-\-x=b=  14 

Then  by  transposing  z,  BC=6 — <c. 

Prob.  6.  The  area  of  a  right  angled  triangle  ABC  (Fig. 
12,)  being  given,  and  the  sides  of  a  parallelogram  inscribed 
in  it,  to  find  the  side  BC. 

Fig.  12.  Fig.  13. 


Prob.  7.  The  three  sides  of  a  right  angled  triangle,  ABC, 
(Fig.  13,)  being  given,  to  find  the  segments  made  by  a  per- 
pendicular, drawn  from  the  right  angle  to  the  hypothenuse. 

The  perpendicular  will  divide  the  original  triangle,  into 
two  right  angled  triangles,  BCD  and  ABD.  (Euc.  8.  6.) 

Prob.  8.  Having  the  area  of  a 
parallelogram  DEFG  (Fig.  14,)  in- 
scribed in  a  given  triangle  ABC, 
to  find  the  sides  of  the  parallelo- 
gram. 

Draw  CI  perpendicular  to  AB. 
By  supposition,  DG  is  parallel  to 
AB. 


244 


ALGEBRA. 


[Sect.  XVI. 


Fig.  15. 


Prob.  9.  Through  a  given  point, 
in  a  given  circle,  so  to  draw  a 
right  line,  that  its  parts,  between 
the  point  and  the  periphery,  shall 
have  a  given  difference. 

In  the  circle  AQBR,  (Fig.  15,) 
let  P  be  a  given  point,  in  the  diam- 
eter AB. 


Prob.  10.  If  the  sum  of  two  of  the  sides  of  a  triangle  be 
1155,  the  length  of  a  perpendicular  drawn  from  the  angle 
included  between  these  to  the  third  side  be  300,  and  the  dif- 
ference of  the  segments  made  by  the  perpendicular,  be  495  ; 
what  are  the  lengths  of  the  three  sides  ? 

Prob.  11.  If  the  perimeter  of  a  right  angled  triangle  be 
720,  and  the  perpendicular  falling  from  the  right  angle 
on  the  hypothenuse  be  144 ;  what  are  the  lengths  of  the 
sides  ? 

Prob.  12.  The  difference  between  the  diagonal  of  a  square 
and  one  of  its  sides  being  given,  to  find  the  length  of  the 
sides. 

Prob.  13.  The  base  and  perpendicular  height  of  any  plane 
triangle  being  given,  to  find  the  side  of  a  square  inscribed  in 
the  triangle,  and  standing  on  the  base,  in  the  same  manner 
as  the  parallelogram  DEFG,  on  the  base  AB,  (Fig.  14.) 

Prob.  14.  Two  sides  of  a  triangle,  and  a  line  bisecting  the 
included  angle  being  given  ;  to  find  the  length  of  the  base 
or  third  side,  upon  which  the  bisecting  line  falls. 

Prob.  15.  If  the  hypothenuse  of  a  right  angled  triangle  be 
35,  and  the  side  of  a  square  inscribed  in  it,  in  the  same  man- 
ner as  the  parallelogram  BEDF,  (Fig.  12,)  be  12  ;  what  are 
the  lengths  of  the  other  two  sides  of  the  triangle  ? 


Art.  399.]  APPLICATION  TO  GEOMETRY.  245 

Prob.  16.  The  number  of  feet  in  the  perimeter  of  a  right 
angled  triangle,  is  equal  to  the  number  of  square  feet  in  the 
area  ;  and  the  base  is  to  the  perpendicular  as  4  to  3.  Re- 
quired the  length  of  each  of  the  sides. 

Prob.  17.  A  grass  plat  12  rods  by  18,  is  surrounded  by  a 
gravel  walk  of  uniform  breadth,  whose  area  is  equal  to  that 
of  the  grass  plat.  What  is  the  breadth  of  the  gravel  walk  ? 

Prob.  18.  The  sides  of  a  rectangular  field  are  in  the  ratio 
of  6  to  5  ;  and  one  sixth  of  the  area  is  125  square  rods. 
What  are  the  lengths  of  the  sides  ? 

Prob.  19.  There  is  a  right  angled  triangle,  the  area  of 
which  is  to  the  area  of  a  given  parallelogram  as  5  to  8.  The 
shorter  side  of  each  is  60  rods,  and  the  other  side  of  the  tri- 
angle adjacent  to  the  right  angle,  is  equal  to  the  diagonal  of 
the  parallelogram.  Required  the  area  of  each  ? 

Prob.  20.  There  are  two  rectangular  vats,  the  greater  of 
which  contains  20  cubic  feet  more  than  the  other.  Their 
capacities  are  in  the  ratio  of  4  to  5  ;  and  their  bases  are 
squares,  a  side  of  each  of  which  is  equal  to  the  depth  of  the 
other  vat.  Required  the  depth  of  each  ? 

Prob.  21.  Given  the  lengths  of  three  perpendiculars,  drawn 
from  a  certain  point  in  an  equilateral  triangle,  to  the  three 
sides,  to  find  the  lengths  of  the  sides. 

Prob.  22.  A  square  public  green  is  surrounded  by  a  street 
of  uniform  breadth.  The  side  of  the  square  is  3  rods  less 
than  9  times  the  breadth  of  the  street ;  and  the  number  of 
square  rods  in  the  street,  exceeds  the  number  of  rods  in  the 
perimeter  of  the  square  by  228.  What  is  the  area  of  the 
square  ? 

Prob.  23.  Given  the  lengths  of  two  lines  drawn  from  the 
acute  angles  of  a  right  angled  triangle,  to  the  middle  of  the 
opposite  sides  :  to  find  the  lengths  of  the  sides. 
21* 


246  ALGEBRA. 


MISCELLANEOUS   PROBLEMS. 

-  ^  •        i      *          ,    '. ••-•          .   •      '  •"-'     .  V1'       V .     - 


1.  WHAT  two  numbers  are  those  whose  difference  is  10; 
and  if  15  be  added  to  their  sum,  the  amount  will  be  43  ? 

2.  There  are  two  numbers  whose  difference  is  14 ;  and  if 
9  times  the  less  be  subtracted  from  6  times  the  greater,  the 
remainder  will  be  33.     What  are  the  numbers  ? 

3.  What  number  is  that  to  which  if  20  be  added,  and  from 
f  of  this  sum,  12  be  subtracted,  the  remainder  will  be  10  ? 

4.  What  number  is  that,  J,  J  and  f  of  which  being  added 
together  will  make  73  ? 

5.  A  and  B  lay  out  equal  sums  of  money  in  trade  ;  A  gains 
,£120,  and  B  loses  c€80 ;  and  now  A's  money  is  triple  that 
of  B.     What  sum  had  each  at  first  ? 

;     6.  What  number  is  that,  J  of  which  exceeds  J  by  72  ? 

7.  There  are  two  numbers  whose  sum  is  37 ;  and  if  3  times 
the  less  be  subtracted  from  4  times  the  greater,  and  the  re- 
mainder be  divided  by  6,  the  quotient  will  be  6.     What  are 
the  numbers  ? 

8.  A  man  has  two  children,  to  J  of  the  sum  of  whose  ages 
if  13  be  added,  the  amount  will  be  17 ;  but  if  from  half  the 
difference  of  their  ages  1  be  subtracted,  the  remainder  will 
be  2.     What  is  the  age  of  each  ? 

9.  A  messenger  being  sent  on  business,  goes  at  the  rate  of 
6  miles  an  hour ;  8  hours  afterwards,  another  is  dispatched 
with  countermanding  orders,  and  goes  at  the  rate  of  10  miles 
an  hour.     How  long  will  it  take  the  latter  to  overtake  the 
former? 


MISCELLANEOUS    PROBLEMS.  247 

10.  To  find  two  numbers  in  the  proportion  of  2  to  3  whose 
product  shall  be  54. 

11.  A  man  agreed  to  give  a  laborer  12s.  a  day  for  every 
day  he  worked,  but  for  every  day  he  was  idle  he  should  for- 
feit Ss.     After  390  days  they  settled,  and  their  account  was 
even.     How  many  days  did  he  work. 

12.  To  find  a  number  to  the  sum  of  whose  digits  if  7  be 
added,   the  result  will  be  3  times  the  left  hand  digit ;  and  if 
from  the  number  itself  18  be  taken,  the  digits  will  be  inverted. 

13.  A  merchant  has  two  kinds  of  tea,  one  of  which  is  worth 
9s.  6d.  per  pound,  the  other  13s.  6d.     How  many  pounds  of 
each  must  he  take  to  form  a  chest  of  104  Ibs.  which  will  be 
worth  ,£56  ? 

14.  To  find  a  number  consisting  of  two  digits,  the  sum  of 
which  is  5 ;  and  if  9  be  added  to  the  number  itself,  the  digits 
will  be  inverted. 

15.  There  is  a  certain  fraction  such,  that  if  you  add  1  to 
its  numerator,  it  becomes  J ;  but  if  ^ou  add  3  to  its  denomi- 
nator, it  becomes  J.     Required  the  fraction. 

16.  Out  of  a  cask  of  water,  which  had  leaked  away  one 
third,  21  gallons  were  drawn,  and  then  being  gauged,  it  was 
found  to  be  half  full.     How  many  gallons  did  it  hold  ? 

17.  It  is  required  to  find  two  numbers  whose  difference  is 
7,  and  their  sum  33. 

18.  At  a  town  meeting  375  votes  were  cast,  and  the  per- 
son elected  to  office  had  a  majority  of  91.     How  many  votes 
had  each  candidate  ? 

19.  A  post  stands  J  in  the  ground,  J  in  the  water,  and  10 
feet  above  the  water.     What  is  the  whole  length  of  it  ? 

20.  A  young  man  the  first  day  after  his  arrival  at  New 
York,  spent  J  of  his  money,  the  second  day  J,  the  third  day 


248  ALGEBRA. 

£,  and  he  then  had  only  26  dollars  left.     How  much  did  he 
have  at  first. 

21.  A  person  being  asked  his  age,  answered  that  f  of  his 
age  multiplied  by  -^  of  his  age,  would  give  a  product  equal 
to  his  age.     How  many  years  old  was  he  ? 

22.  A  man  leased  a  house  for  99  years  ;   and  being  asked 
how  much  of  the  time  had  expired,  replied  that  two  thirds  of 
the  time  past,   was  equal  to  four  fifths  of  the  time  to  come. 
How  many  years  had  expired  ? 

23.  On  commencing  the  study  of  his  profession,  a  man 
found  that  \  of  his  life  had  been  spent  before  he  learned  his 
letters,  £  at  a  public  school,  £  at  an  academy,  and  4  years  at 
college.     How  old  was  he  ? 

24.  It  is  required  to  find  a  number  such,  that  whether  it  be 
divided  into  two,  or  three  equal  parts,  the  product  of  its  parts 
will  be  equal. 

25.  Two  persons  154  miles  apart,  set  out  at  the  same  time 
to  meet  each  other,  one  travelling  at  the  rate  of  3  miles  in 
2  hours,  the   other  5  miles  in  4  hours.     How  long  before 
they  meet  ? 

26.  A  man  and  his  wife  usually  drank  a  cask  of  beer  in 
12  days,  but  when  the  man  was  absent,  it  lasted  the  lady  30 
days.     How  long  would  it  last  the  man,  if  his  wife  were 
absent  ? 

27.  A  shepherd   being   asked  how  many  sheep   he  had, 
replied  if  he  had  as  many  more,  half  as  many  more,  and  7J 
sheep,  he  would  then  have  500.     How  many  had  he  ? 

28.  A  farmer  hired  two  men  to  do  a  job  of  work  for  him  ; 
one  could  do  the  work  in  10  days,  the  other  in  15.     How 
long  would  it  take  both  together  to  do  {ne  same  work  ? 

29.  A  scaffold  of  hay  will  keep  5  horses  or  7  oxen,  87 
days.     How  long  will  it  keep  2  horses  and  3  oxen  ? 


MISCELLANEOUS  PROBLEMS.  249 

30.  A  and  B  together  can  build  a  boat  in  20  days  ;  with 
the  assistance  of  C,  they  can  do  it  in  12  days.     How  long 
would  it  take  C  to  build  the  boat  ? 

31.  There  is  a  cistern  with  two  aqueducts  ;   one  will  fill  it 
in  30  minutes,  the  other  will  empty  it  in  40.     How  long  will 
it  take  to  fill  it,  if  both  run  together  ? 

32.  Required  to  divide  1  shilling  into  pence  and  farthings 
in  such  a  proportion  that  there  may  be  39  pieces. 

33.  A  man  divided  a  small  sum  of  money  among  his  chil- 
dren in  the  following  manner,  viz.  to  the  first  he  gave  £  of 
the  whole  -f-4  pence,  to  the  second  £  of  the  remainder  -f-8 
pence,  to  the  third  £  of  the  remainder  -f-12  pence,  and  so  on, 
giving  to  all  an  equal  sum  till  he   had  distributed  the  whole. 
Required  the  number  of  shares  and  the  sum  distributed. 

34.  A  hare  has  50  leaps  the  start  of  a  hound,  and  takes  4 
leaps  while  the  hound  takes  3  ;   but  2  leaps  of  the  hound  are 
equal  to  3  of  the  hare.     How  many  leaps  will  the  hound 
take  in  catching  the  hare  ? 

35.  A  cistern  holding  43  gallons,  is  to  be  filled  in  12  minutes 
by  2  pipes  running  alternately.     The  first  runs  4  gallons  a 
minute,  and  the  second  3  gallons  a  minute.     How  long  did 
each  run  ? 

36.  A  and  B  start  at  the  same  time  and  place  to  go  round 
an  island  600  miles  in  circumference.     A  goes  30  miles  a 
day,  and  B  20.     How  long  before  they  will  both  be  at  the 
starting  point  together,  and  how  far  will  each  have  travelled  ? 

37.  A  has  ,£100,  B  .£48.     A  robber  takes  twice  as  much 
from  A,  as  from  B.     A  now  has  3  times  as  much  as  B. 
What  was  taken  from  each  ? 

38.  It  is  required  to  divide  1200  dols.   between  A,  B  and 
C ;    B  has  256+J  of  A's  share  ;    C  has  270+J  of  B's. 
What  was  the  share  of  each  ? 


250  ALGEBRA. 

39.  There  are  3  pieces  of  cloth  of  different  value.     The 
average  price  of  the  first  and  second  is  7  dols.  per  yard,  that 
of  the  second  and  third  is  9  dols.,  and  the  average  price  of  all 
is  |  of  the  third.     What  are  the  several  prices  ? 

40.  A  pipe  will  fill  a  cistern  in  11  hours.     After  running 
5  hours,  another  is  opened,  and  then  the  two  fill  it  in  2  hours. 
In  what  time  would  the  last  fill  it  ? 

41.  A  man  bought  a  cask  of  wine  and  ^  of  it  leaking  out, 
he  sold  the  rest  at  $2.50  per  gallon  and  neither  gained  nor 
lost  by  his  bargain.     What  did  he  give  per  gallon  for  his  wine  ? 

42.  A  and  B  start  at  the  same  time  and  in  the  same  direc- 
tion, but  directly  opposite  each  other,  to  go  round  a  circular 
pond  536  yards  in  circumference  ;  A  goes  11  yards  a  minute 
and  B  34  in  3  minutes.     In  how  long  time  will  B  overtake  A  ? 

43.  A  pipe  fills  |  of  a  cistern  in  1  hour ;  2  hours  after  an- 
other is  opened  and  would  have  hastened  the  filling  of  it  1 
hour ;  but  2  hours  after,  a  third  begins  to  discharge,  and  the 
cistern  is  finally  filled  in  the  time  the  first  would  have  filled 
it.     Required  the  time  of  the  second  in  filling  it,  or  the  third 
in  emptying  it  ? 

44.  A  young  man  commencing  business  with  a  determina- 
tion to  become  rich,  supported  himself  for  $500  a  year,  and 
at  the  close  of  every  year  increased  his  property  by  a  third 
part  of  what  remained  after  his  expenses  were  deducted. 
In  five  years  he  was  worth  $104,400.     What  was  his  original 
stock  ? 

45.  At  noon  the  hour  and  minute  hand  of  a  clock  are 
together.     How  soon  will  they  be  together  again  ? 

46.  A  gentleman  bought  5  bushels  of  wheat  and  6  of  corn 
for  27s. ;  he  afterwards  bought  4  bushels  of  wheat  and  3  of 
corn  for  18s.     What  did  he  pay  per  bushel  for  each  ? 


MISCELLANEOUS    PROBLEMS.  251 

47.  A  farmer  hired  4  men  and  8  boys  for  a  week,  and 
paid  them  $40 ;    the   next  week  he  hired  7  men  and  6  boys 
for  $50.     How  much  did  he  pay  each  by  the  week  ? 

48.  Divide  72  into  four  such  parts  that  the   first  increased 
by  5,  the  second  diminished  by  5,  the  third  multiplied  by  5, 
and  the  fourth  divided  by  5,  the  sum,  difference^  product  and 
quotient,  will  be  equal  ? 

49.  A  and  B  can  print  a  certain  work  in  8  days,  A  and  C 
in  9  days,  'B  and  C  in  10  days.     How  long  would  it  take 
each  one  alone  to  do  the  work  ? 

50.  Three  boys  were  playing  marbles  ;    the  first  game,  A 
loses  to  B  and  C,  as  many  as  each  of  them  had  at  the  begin- 
ning ;    next,  B  loses  to  A  and  C,  as   many  as  each  of  them 
had  at  the  end  of  the   first  game ;    last,  C  loses  to  A  and 
B,  as  many  as  each  of  them  had  at  the  end  of  the  second 
game.     Each  then  had  16  marbles.     How  many  had  each 
at  first  ? 

51.  It  is  required  to  find  two  numbers  whose  product  shall 
be  54,  and  the  difference  of  their  squares  45. 

52.  Required  the  side  of  a  rectangular  field  which  contains 
1584  square  rods,  and  its  length  exceeds  its  breadth  by  8  rods. 

53.  The  united  ages  of  a  man  and  his  wife  were  42  years, 
and  the   product  of  their  ages  432.     What  was  the  age  of 
each  ? 

54.  A  single  lady  being  asked  her  age,  considered  the 
question   impertinent  and   gave  an  evasive  answer,   saying, 
*'  if  you  take  4  years  from   my  age,  and  extract  the  square 
root  of  the  remainder,  and  multiply  the  root  by  4,  and  add  4 
to  the  product,  the  sum  will  be  24.     What  was  her  age  ? 

55.  A  peach  orchard  which  contained  900   trees  was  so 
planted,  that  there  were  11  rows  more  than  there  were  trees 


252  ALGEBRA. 

in  a  row.     How  many  rows  were  there,  and  how  many  trees 
in  each  row  ? 

56.  A  man  purchased  a  tract  of  land  in  a  square  form, 
which  contained  as  many  acres  as  there  were  rails  in   the 
fence  with  which  it  was  inclosed  ;  the  rails  were  1 1  feet  long 
and  the  fence  was  4  rails  high.     How  many  acres  did  the 
tract  contain  ? 

57.  A  man  bought  80  pounds  of  pepper  and  36  pounds  of 
saffron,  so  that  for  8  crowns  he  had  14  pounds  of  pepper 
more  than  of  saffron  for  26  crowns  ;   and  the  amount  he  laid 
out  was  188  crowns.     How  many  pounds  of  pepper  did  he 
buy  for  8  crowns  ? 

58.  It  is  required  to  find  four  numbers  in  arithmetical  pro- 
gression such,  that  the  product  of  the  extremes  shall  be  45, 
and  the  product  of  the  means  77. 

59.  It  is  required  to  find  three  numbers  in  geometrical 
progression  such,  that  their  sum  shall  be  14,  and  the  sum  of 
their  squares  84. 

60.  The  hypothenuse  of  a  right  angled  triangle  is  13  feet, 
and   the  difference  between  the  other  two  sides  is  7.     Re* 
quired  the  sides. 

61.  The  perpendicular  of  a  plane  triangle  is  300  feet ;  the 
sum  of  two  of  the  sides  is  1150  feet,  and  the  difference  of  the 
segments  of  the  base  is  495  feet.     Required  the   base  and 
the  sides. 

62.  In  a  plane  triangle   the  base  is  50  feet,  the  area  796 
feet,  and  the  difference  of  the  sides  is  10  feet.     Required  the 
sides  and  perpendicular. 

THE     END. 


031 


